diff --git "a/79E0T4oBgHgl3EQfwQGR/content/tmp_files/load_file.txt" "b/79E0T4oBgHgl3EQfwQGR/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/79E0T4oBgHgl3EQfwQGR/content/tmp_files/load_file.txt" @@ -0,0 +1,791 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf,len=790 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='02630v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='AG] 15 Aug 2022 HEIGHT PAIRING AND NEARBY CYCLES A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Beilinson To Yuri Ivanovich Manin with deepest gratitude Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' We prove that, as was conjectured by Spencer Bloch, the Hodge period of some limit Hodge structures equals the height pairing of algebraic cycles on the resolution of singularities of the singular fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Introduction: the theorem and the idea of the proof 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The Hodge period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Suppose we have a Q-Hodge structure E with weights in [−2, 0] equiped with isomorphisms ι0 : grW 0 E = Q(0), ι−2 : grW −2E = Q(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' One defines the Hodge period ⟨E⟩ = ⟨E, ι0, ι−2⟩ ∈ R as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Consider the R-Hodge structure E ⊗ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Since the weight filtration on any R-Hodge structure with two consequitive weights (canonically) splits one has E ⊗ R = G ⊕ grW −1E ⊗ R where G is an extension of R(0) by R(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Our ⟨E⟩ is the class of this extension in Ext1(R(0), R(1)) = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' One computes ⟨E⟩ explicitly as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Let ER be E ⊗ R viewed a plain R-vector space, EC be its complexification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Let 1F 0 ∈ F 0 ⊂ EC be any lifting of ι−1 0 (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Then ⟨E⟩ is the image of 1F 0 in (ER + W−1EC)/(ER + (F 0 ∩ W−1EC)) ∼ ← W−2EC/W−2ER = C/2πiR ∼ ← R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' A geometric example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Let Y be a smooth proper equidimensional algebraic variety over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' We denote by Hi(Y ) the homology of Hi(Y (C), Q) seen as an object of the category of Q-Hodge structures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' ditto for relative homology, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Let Zm(Y ) be the group of algebraic m-cycles on Y with Q-coefficients, Zm(Y )0 := Ker(cl : Zm(Y ) → H2m(Y )(−m)) be the subgroup of cycles homologically equivalent to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' For a closed subset P ⊂ Y let Zm(P) ⊂ Zm(Y ) be the subgroup of cycles supported on P, Zm(P)0 := Zm(P) ∩ Zm(Y )0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' For an m-cycle A on Y we denote by |A| its support (which is a closed subset of Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Suppose m + m′ = dim Y − 1 and we have A ∈ Zm(Y )0, B ∈ Zm′(Y )0 such that |A| ∩ |B| = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Set E|A|,|B| := H2m+1(Y ∖ |B|, |A|)(−m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Notice that E|B|,|A| = E∗ |A|,|B|(1) by the Poicar´e duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' E|A|,|B| has weights in [−2, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' One has grW −2E|A|,|B| = Zm′(|B|)∗ 0(1), grW −1E|A|,|B| = H2m+1(Y )(−m), grW 0 E|A|,|B| = Zm(|A|)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Notice that H2m(|A|)(−m) = Zm(|A|) and H>2m(|A|) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' By the Poincar´e duality Hi(Y, Y ∖|B|)(− dim Y ) = H2 dim Y −i(|B|)∗, hence H2m+2(Y, Y ∖|B|)(−m) = 1991 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Primary 14C25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Secondary 14D07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' height pairing, nearby cycles, Hodge periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Typeset by AMS-TEX 1 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' BEILINSON (H2m′(|B|)(−m′))∗(1) and H<2m+2(Y, Y ∖ |B|) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Now use the long exact ho- mology sequences for (Y ∖ |B|, |A|) and (Y, Y ∖ |B|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' □ Denote by EA,B the Hodge structure obtained from H|A|,|B| by pullback by A and pushforward by B: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1) Zn(|B|)∗ 0(1) ֒→ E|A|,|B| ։ Zm(|A|)0 B ↓ ↑ A Q(1) ֒→ EA,B ։ Q(0) Our EA,B is as in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1, so we have ⟨EA,B⟩ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The height pairing (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' [B], [Bl1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Let k be a subfield of C and suppose that Y comes from a variety Yk over k, Y = Yk ⊗ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Let Zm(Yk) ⊂ Zm(Y ) be the group of algebraic cycles with Q-coefficients on Yk, Zm(Yk)0 := Zm(Yk) ∩ Zm(Y )0, and let CHm(Yk)0 ⊂ CHm(Yk) be their quotients modulo the rational equivalence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' One checks (see §2) that if A, B as above are cycles on Yk then the class of ⟨EA,B⟩ in R/Q log |k×| depends only on linear equivalence classes of A and B, and so one has a bilinear height pairing (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1) ⟨ , ⟩Yk : CHm(Yk)0 ⊗ CHm′(Yk)0 → R/Q log |k×|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Namely ⟨a, b⟩Yk = ⟨EA,B⟩ where A, B are any cycles on Yk of classes a, b such that |A| ∩ |B| = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' If k = Q and we assume some motivic rationality conjectures (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='3) of [B]) then ⟨EA,B⟩ can be corrected (by adding a finite sum of corrections log(p)⟨EA,B⟩p where p is a prime, ⟨EA,B⟩p is defined using the Gal(Qp)-action on EA,B ⊗ Qℓ) so that the resulting real number depends only on rational equivalence classes of A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' In this manner (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1) lifts naturally to an R-valued pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Finding elements of Chow groups that are homologically equivalent to zero is an art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Spencer Bloch described one situation where they naturally arise, and conjectured that the height pairing of his cycles can be computed in s different way, namely, as Hodge periods of some nearby cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' We start with preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Let X be a smooth variety over C of pure dimension n ≥ 2, S be a smooth curve, 0 ∈ S be a closed point, and f : X → S be a proper map which is smooth otside a finite subset {xα} of the fiber X0 = f −1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Let Zα be the projectivized tangent cone to X0 at xα;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' this is a hypersurface in the projectivization Pα := P(TxαX) of the tangent space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' denote by dα its degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' We assume the next condition: (∗) All hypersurfaces Zα are smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Let π : Y → X0 be the blowup of X0 at {xα}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Condition (∗) implies that Y is a smooth variety, and Zα are pairwise disjoint divisors on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Set Z := ⊔Zα and K := Ker(Hn−2(Z) → Hn−2(Y )) = Im(Hn−1(Y, Z) → Hn−2(Z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' If n = 2 then let K0 ⊂ K be the subgroup of those elements A = ΣAα that deg Aα = 0 for every α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' One has a natural map Hn−1(X0) → Hn−1(Y, Z) defined as the composition Hn−1(X0) → Hn−1(X0, {xα}) ∼ ← Hn−1(Y, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' (i) The map Hn−1(X0) → Hn−1(Y, Z) is an isomorphism if n > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' If n = 2 it is injective and its image equals the preimage of K0 in Hn−1(Y, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' (ii) Hn−1(Y, Z) has weights 1 − n and 2 − n, and grW 2−nHn−1(Y, Z) = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The map HEIGHT PAIRING AND NEARBY CYCLES 3 Hn−1(Y ) → Hn−1(Y, Z) has image W1−nHn−1(Y, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' If n is even then Hn−1(Y ) ∼ → W1−nHn−1(Y, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' (i) Replace Hn−1(Y, Z) by Hn−1(X0, {xα}) and use the long exact homology sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' (ii) The first assertions follow from the exact homology sequence and purity of weights on H·(Y ), H·(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The last one comes because Hn−1(Z) = 0 if n is even (since Zα are hypersurfaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' □ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Consider a variation of Q-Hodge structures V on S ∖ {0} with fibers Vs = Hn−1(Xs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' One has a nondegenerate intersection pairing ( , ) : V ⊗ V → Q(n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Choose a parameter t at 0 ∈ S and consider the limiting (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' nearby cycles) Hodge structure ψtV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Let ψun t V be its direct summand where the monodromy acts unipotently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Since ψun t commutes with duality, ( , ) yields self-duality pairing on it that we denote again by ( , ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' One has the log of monodromy morphism N = NV : ψun t V(1) → ψun t V and the specialization morphism sp : ψun t V → Hn−1(X0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Let (ψun t V)N := Coker(NV) be the monodromy coinvariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The next assertion follows from the local invariant cycles theorem, see 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='5 for a detailed proof: Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' sp factors through the isomorphism (ψun t V)N ∼ → Hn−1(X0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' ψun t V has weights in [−n, 2 − n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' One has grW 2−nψun t V = K if n > 2 and grW 2−nψun t V = K0 if n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' By self-duality, grW −nψun t V = (grW 2−nψun t V)∗(n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' If n is even then grW 1−nψun t V = Hn−1(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Since ψun t V is self-dual and N is nilpotent, the claim follows from the propo- sition and the lemma in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' □ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Bloch cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' We are in the setting of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' suppose n is even, n = 2m + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Let A = ΣAα be an m-cycle on Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' We say that A is a Bloch cycle if it is homologically equivalent to zero on Y , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=', cl(A) lies in K(−m) ⊂ Hn−2(Z)(−m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' If m = 0 then we demand, in addition, that cl(A) ∈ K0 ⊂ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' If A is a Bloch cycle then each cl(Aα) ∈ Hn−2(Zα)(−m) is primitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The composition Hn−2(Z)(−m) → Hn−2(Y )(−m) → Hn−4(Zα)(−m + 1), where the second arrow is the pullback by Zα ֒→ Y , sends any class c = Σcα to cα ∩ c1(O(−1)) (for O(−1) is the normal bundle to Zα in Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' This composition kills cl(Aα) since the first arrow does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' □ If A, B are two Bloch cycles then we denote by Eψ A,B = Eψ A,B,t the Hodge struc- ture obtained from ψun t V(−m) by pullback by cl(A) and pushforward by cl(B)∗: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1) K∗(m + 1) → ψun t V(−m) → K(−m) cl(B)∗ ↓ ↑ cl(A) Q(1) ֒→ Eψ A,B ։ Q(0) Our Eψ A,B is as in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1 so we have ⟨Eψ A,B⟩ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Consider the case when we have single singular point x0 ∈ X0 of f and the singularity at x0 is quadratic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Then the monodromy action on ψtV is unipotent, the only possible Bloch cycle is the difference A of the rulings of the quadric Z0, and it is actually a Bloch cycle if and only if the monodromy action on ψtV is nontrivial or, equivalently, the Hodge structure on Hn−1(X0) is not pure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' BEILINSON Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' (i) If m = 0 then the curve X0 can have either 1 or 2 irreducible compo- nents, and A is a Bloch cycle if and only if X0 is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' (ii) If X/S is a family of quadratic hypersurfaces in Pn then A is not a Bloch cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' (iii) If X/S is a family of hypersurfaces of degree d on a given smooth projective variety P then A is a Bloch cycle if d is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' (i) is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' (ii) follows since the global monodromy for quadratic hypersur- faces is ±1, and so it can’t contain non-trivial unipotent local monodromy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' (iii) Consider the corresponding map r : S → B := {hypersurfaces of degree d on P}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Since X is smooth r is transversal to the locus D ⊂ B of degenerate hypersurfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Replacing S by a germ of another transversal to D that intersects D near r(0) would not change the topology of X over a small disc around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' So we can assume that S is a Zariski open subset of the base of a Lefschetz pencil on P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Then, since local monodromies of a Lefschetz pencil are all conjugate, triviality of one local monodromy amounts to triviality of the global monodromy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Thus A is a Bloch cycle if and only if the global monodromy on V is not trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Let us check that this happens for large enough d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' If R ⊂ P is the axis of our pencil then H·(X) = H·(P) ⊕ H·−2(R)(−1), and so hn−1,0(P) = hn−1,0(X) which equals hn−1,0(Xs) if the global monodromy is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Thus the monodromy is not trivial when hn−1,0(Xs) > hn−1,0(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' To finish the argument it remains to notice that hn−1,0(Xs) ≥ dim(H0(P, Ωn P (d))/H0(P, Ωn P )), and so it tends to ∞ when d → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' □ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Statement of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Now suppose we have a subfield k ⊂ C and our datum is defined over k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=', there is Xk/Sk, a closed point 0 of Sk, a parameter t on Sk at 0, and Bloch cycles A, B on Zk such that X/S, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=', come by base change k → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Let a ∈ CHm(Yk)0, b ∈ CHm(Yk)0 be the classes of A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The next result was conjectured by Spencer Bloch: Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' One has ⟨a, b⟩Yk = ⟨Eψ A,B⟩ mod Q log |k×|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' In case n = 1 the theorem was proven in [BlJS].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Suppose we are in the situation of Remark in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' If ⟨Eψ A,B⟩ is corrected in the same way as was discussed there, then the theorem lifts to an equality of real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The proof does not change;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' we will not discuss it below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Reformulation of the theorem that discards Hodge periods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' the idea of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Let A′, B′ be cycles on Yk of classes a, b such that |A′| ∩ |B′| = ∅ (no- tice that they are, most probably, not supported on Zk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' We want to show that ⟨EA′,B′⟩ = ⟨Eψ A,B⟩ (see 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Let us compare the Hodge structures E = EA′,B′ and Eψ = Eψ A,B themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Their weights lie in [−2, 0], and one has a canonical identification grW E = grW Eψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Indeed, grW 0 E(ψ) = Q(0), grW −2E(ψ) = Q(1) by the constructions, and grW −1E = H2m+1(Y )(−m) = grW −1E(ψ) by the lemma in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2, and the one in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='4 combined with the corollary in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' This identification lifts (uniquely) to W−1E = W−1Eψ and E/W−2E = Eψ/W−2Eψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Indeed, the classes of extensions 0 → H2m+1(Y )(−m) → E(ψ)/W−2E(ψ) → Q(0) → 0 both equal Deligne cohomol- ogy class clD(A) (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Griffiths’ Abel-Jacobi periods) of A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' by duality, the classes of (the duals to) extensions 0 → Q(1) → W−1E(ψ) → H2m+1(Y )(−m) → 0 both equal to clD(B) (see loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' HEIGHT PAIRING AND NEARBY CYCLES 5 Now suppose we have a Q-Hodge structure H of weight −1 and two classes a ∈ Ext1(Q(0), H), b ∈ Ext1(H, Q(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Consider the set EH a,b = EH(H)a,b of all Hodge structures E with weights in [−2, 0] and equipped with identifications grW 0 E = Q(0), grW −1E = H, grW −2E = Q(1) such that the extensions E/W−2E and W−1E have classes a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The group Ext1(Q(0), Q(1)) = C× ⊗ Q acts on EH a,b by the Baer sum action, and EH a,b is a C× ⊗ Q-torsor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Notice that for q ∈ C× one has ⟨q · E⟩ = log |q| + ⟨E⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Applying this format to H = H2m+1(Y )(−m), a = clD(A), b = clD(B) and EA′,B′, Eψ A,B ∈ EH a,b we get EA′,B′ − Eψ A,B ∈ C× ⊗ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Now the theorem in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='8 follows immediately from the next result (notice that the Hodge periods and the height pairing play no role here): Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' One has EA′,B′ − Eψ A,B ∈ k× ⊗ Q ⊂ C× ⊗ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The theorem would be an immediate corollary of the motivic formalism if all the above constructions could be spelled in motivic world: Indeed, we would have then a motivic version EM of EH which is an Ext1 M(Q(0), Q(1)) = k× ⊗ Q-torsor equipped with the Hodge realization embedding EM ֒→ EH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' our EA′,B′, Eψ A,B would come from elements of EM, and so their difference lies in k× ⊗ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The only problem is that in the present day formalism of motives, due to Voevodsky, Ayoub, and Cisinski-D´eglise, the t-structure is not available, so we do not have the motivic version of separate homology groups like Hi(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The actual proof is an exercise in spelling out the constructions in a way that makes the t-structure redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' I am very grateful to Spencer Bloch for explaining me his conjecture and stimu- lating discussions (pity Spencer refused to coauthor the article), to Volodya Drinfeld for valuable comments and discussions, and to Luc Illusie for calling my attention to the construction of [I] which helped to clearify and simplify the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The height pairing and the construction of EM a,b ∈ EM a,b ⊂ EH a,b This section is a variation on the theme of [Bl2] and [G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Let C be a stable dg category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' It yields two other dg categories C(1) and C(2) constructed as follows: An object of C(1) is a closed morphism α : M → N of degree 0 in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' One has Hom((M, N, α), (M ′, N ′, α′))i = Hom(M, M ′)i ×Hom(N, N ′)i ×Hom(M, N ′)i+1 ⊂ Hom(Cone(α), Cone(α′))i, and the differential is defined so that the latter embed- ding is a morphism of complexes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' the composition of morphisms is defined in a sim- ilar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' There are three dg functors C(1) → C which send (M, N, α) to M, N, and Cone(α) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' We can view C(1) as the category of distinguished triangles, and the rotation yields an autoequivalence ρ : C(1) → C(1) which sends α : M → N to ρ(α) : N → Cone(α);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' the inverse autoequivalence is ρ−1(α) : Cone(α)[−1] → M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' An object of C(2) is a datum (P, M, Q, α, β, κ) where P, M, Q are objects of C, α ∈ Hom(P, M)1, β ∈ Hom(M, Q)1 are closed maps, and κ ∈ Hom(P, Q)1 is such that d(κ) = βα;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' we sometimes abbreviate it to (α, β, κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' One can assign to such a datum an object E = E(α, β, κ) ∈ C which equals P ⊕ M ⊕ Q with α, β, and −κ added as the components to the differential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1 There is a filtration Q ⊂ Cone(β : M[−1], Q) ⊂ E, and morphisms in C(2) are the same as morphisms between the 1Thus E = Cone((α, κ) : P [−1] → Cone(β : M[−1] �� Q)) = Cone((κ, β) : Cone(α : P [−2] → M[−1]) → Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' 6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' BEILINSON corresponding objects E that preserve this filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' We have two dg functors C(2) → C(1) which send to (α, β, κ) to α : P[−1] → M and β : M → Q[1], and six dg functors C(2) → C which send (α, β, κ) to P, M, Q, Cone(α : P[−1] → M), Cone(β : M[−1] → Q), and E(α, β, κ) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The dg category C(3) carries a natural involution σ which sends (P, M, Q, α, β, κ) to the object (Q[−1], E(α, β, κ), P[1], ασ, βσ, 0) where ασ and βσ are the evident embedding and projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' One can view an object (α, β, κ) ∈ C(2) as an object of C equipped with a 3-step filtration in two different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Namely, this could be E(α, β, κ) equipped with an evident filtration with successive quotients Q, M, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Or this could be M equipped with a filtration whose successive quotients are P[−1], E(α, β, κ), and Q[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The involution σ exchanges the two perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' For C as above we denote by C× the ∞-groupoid of its homotopy equivalences, by C×τ the corresponding 1-truncaded plain groupoid, and by HC the homotopy category of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' For S, T ∈ C set Exti(S, T ) := HiHom(S, T ) = HomHC(S, T [i]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Denote by Ext(S, T ) the plain Picard groupoid of extensions that corresponds to the two-term complex τ [0,1]Hom(S, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' For M, N ∈ C let C(1)× M,N be the ∞-groupoid of collections (α′ : M ′ → N ′, ιM, ιN) where (α′ : M ′ → N ′) ∈ C(1) and ιM : M → M ′, ιN : N → N ′ are homotopy equiv- alences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' It is equivalent to the Picard ∞-groupoid that corresponds to the complex τ ≤0Hom(M, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The 1-truncated plain Picard groupoid C(1)×τ M,N corresponds to the two-term complex τ [−1,0]Hom(M, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Similarly, for three objects P, M, Q ∈ C we have the ∞-groupoid C(2)× P,M,Q whose objects are data (P ′, M ′, Q′, α′, β′, κ′, ιP , ιM, ιQ) where (P ′, M ′, Q′, α′, β′, κ′) ∈ C(2) and ιP : P → P ′, ιM : M → M ′, ιQ : Q → Q′ are homotopy equivalences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The 1-truncated plain groupoid C(2)×τ P,M,Q contains a normal subgroup Ext0(P, Q) = HomHC(P, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Let by E = E(M) = E(P, M, Q) be the quotient groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' It is equivalent to the groupoid of triples (α, β, κ) where α ∈ Hom(P, M)1, β ∈ Hom(M, Q)1 are closed maps, and κ ∈ Hom(P, Q)1/d(Hom(P, Q)0) is such that d(κ) = βα;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' a morphism (α, β, κ) → (α′, β′, κ′) in E is a pair (φ, ψ) where φ ∈ Hom(P, M)0/d(Hom(P, M)−1), ψ ∈ Hom(M, Q)0/d(Hom(M, Q)−1) are such that α′ − α = d(φ), β′ − β = d(ψ), κ′ − κ = βφ + ψα + ψd(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The projection C(2) P,M,Q → C(1) P [−1],M × C(1) M,Q[1] yields a map of plain groupoids E(P, M, Q) → C(1)×τ P [−1],M × C(1)×τ M,Q[1] = Ext(P, M) × Ext(M, Q), (α, β, κ) �→ (α, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The group Ext1(P, Q) acts on E by translations of κ, and non-empty fibers Eα,β are Ext1(P, Q)-torsors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' E(P, M, Q) is naturally functorial with respect to P and Q: every pair of closed morphisms µ : P1 → P and ν : Q → Q1 yields a map E(P, M, Q) → E(P1, M, Q1), (α, β, κ) �→ (αµ, νβ, νκµ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' is compatible with the Ext1(P, Q)-action via the map (µ∗, ν∗) : Ext1(P, Q) → Ext1(P1, Q1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Suppose Ext2(P, Q) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Then Eα,β are non-empty, and the addition maps Eα1,β × Eα2,β → Eα1+α2,β, Eα,β1 × Eα,β2 → Eα,β1+β2 define on E the structure of an Ext1(P, Q)-biextension of (Ext(P, M), Ext(M, Q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' In our first example C is the dg category whose homotopy category is the bounded derived category DH of the category H of Q-Hodge structures, and HEIGHT PAIRING AND NEARBY CYCLES 7 P = Q(0), Q = Q(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' We denote the corresponding E by EH = EH(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Then Ext̸=1 DH(P, Q) = 0 and Ext1 DH(P, Q) = C× ⊗ Q, so EH is a C× ⊗ Q-biextension of (Ext(Q(0), M), Ext(M, Q(1))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Let Ext1 0(Q(0), M) ⊂ Ext1(Q(0), M), Ext1 0(M, Q(1)) ⊂ Ext1(M, Q(1)) be the subgroups of those elements a, b that the maps H0a : Q(0) → H1M, H−1b : H−1M → Q(1) vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Let Ext0(Q(0), M) ⊂ Ext(Q(0), M), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=', be the Picard groupoids of such extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Suppose that Hom(Q(0), H0M) = Hom(H0M, Q(1)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' (i) The restriction of EH to (Ext0(Q(0), M), Ext0(M, Q(1))) descends to the C×⊗Q- biextension of (Ext1 0(Q(0), M), Ext1 0(M, Q(1))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' (ii) EH is naturally functorial with respect to M: if ϕ : M → M ′ is a morphism, and we have a′ ∈ Ext1 0(Q(0), M ′), b′ ∈ Ext1 0(M ′, Q(1)) with ϕ∗(a) = a′, ϕ∗(b′) = b then there is a canonical identification EH(M)a,b = EH(M ′)a′,b′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' (iii) The isomorphisms Ext1 0(Q(0), M) ∼ → Ext1(Q(0), H0M), Ext1 0(M, Q(1)) ∼ → Ext1 (H0M, Q(1)) which assign to an extension its zero cohomology, lifts naturally to an isomorphism of biextensions H0 : EH(M) ∼ → EH(H0M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' One has EH0 = H0E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Let us prove (i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' the rest is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' We need to check that for every closed α ∈ Hom1 0(Q(0), M), β ∈ Hom1 0(M, Q(1)) the action of Aut(α)×Aut(β) = Hom(Q(0), M) ×Hom(M, Q(1)) on EH α,β is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Since H has homological dimension 1 our M is isomorphic to the direct sum of its homologies and so Aut(α) = Ext1(Q(0), H−1M), Aut(β) = Ext1(H1(M), Q(1)) by the condition on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The action of (e, h) ∈ Ext1(Q(0), H−1M)×Ext1(H1(M), Q(1)) on EH α,β is the translation by H−1(β)e + hH0(α) which is 0 since α, β ∈ Ext1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Suppose that H0M is pure of weight −1 (which implies the condition of the lemma in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Then the function EH(M) → R, (α, β, κ) �→ ⟨E(α, β, κ)⟩ := ⟨H0E(α, β, κ)⟩, see 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1, is a natural trivialization of the R-biextension log |EH(M)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Everything said in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='3 works for the category HR of R-Hodge structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The extension of scalars functor H → HR, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' �→?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' ⊗ R, yields a morphism of our biex- tensions EH(M) → EHR(M ⊗ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The map Ext1(Q(0), Q(1)) → Ext1(R(0), R(1)) equals log | | after the standard identifications of the Ext groups with, respectively, C× ⊗ Q and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Since Ext1(R(0), H0M ⊗ R) = Ext1(H0M ⊗ R, R(1)) = 0 by the condition on M, one has EHR(M ⊗ R) = EHR(H0M ⊗ R) = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The map EH(M) → EHR(M ⊗ R) = R is ⟨ ⟩ of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Let k ⊂ C be a subfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Denote by DM(k) the dg category of geometric Voevodsky Q-motives over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' We have the Hodge realization dg functor DM(k) → DH, M �→ M H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Consider the story of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2 for C = DM(k) with P = Q(0), Q = Q(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' As before one has Ext̸=1 DM(k)(Q(0), Q(1)) = 0, and there is a canonical identification Ext1(Q(0), Q(1)) = k× ⊗ Q such that the Hodge realization map between the Ext1’s is the embedding k× ⊗ Q ֒→ C× ⊗ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' So for any M ∈ DM(k) we get a k× ⊗ Q-biextension of (Ext1(Q(0), M), Ext1(M, Q(1))) together with the Hodge realization morphism EM(M) → EH(M) := EH(M H) of the biextensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Since the homomorphism k× ⊗ Q ֒→ C× ⊗ Q is injective, the maps of torsors EM(M)α,β → EH(M)α,β := EH(M)αH,βH are injective too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' We define Ext1 0(Q(0), M) ⊂ Ext1 0(Q(0), M) and Ext1 0(M, Q(1)) ⊂ Ext1(M, Q(1)) as preimages of the Ext1 0 subgroups of the Hodge setting by the Hodge realiza- 8 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' BEILINSON tion maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Assume that H0M H is pure of weight −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Then (i) and (ii) of the lemma in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='3 remain true in the DM(k) setting (with C× replaced by k×): this follows from loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' by Remark above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Thus we have a k× ⊗ Q-biextension EM(M) of (Ext1 0(Q(0), M), Ext1 0(M, Q(1))) together with a map of biextensions EM(M) → EH(M), so the lemma in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='4 provides a natural trivialization of the R-biextension log |EM(M)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The image of EM a,b in R/Q log |k×| depends only on a, b ∈ Ext1 0(M, Q(1)) × Ext1 0(Q(0), M), and we denote it by ⟨a, b⟩M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' It is clearly biadditive with respect to a, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2 We have defined a canonical height pairing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1) ⟨ ⟩M : Ext1 0(Q(0), M) × Ext1 0(M, Q(1)) → R/Q log |k×|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' We return to the situation of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='3 and set M := M(Yk)(−m)[−1 − 2m] where M(Yk) is the motive of Yk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' One has Ext1(Q(0), M) = CHm(Yk), Ext1(M, Q(1)) = CHm′(Yk) by the Poincar´e duality, and Ext1 0 are the subgroups CH(Yk)0 of cycles homologically equivalent to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Therefore we get a k× ⊗ Q-biextension EM of (CHm(Yk)0, CHm′(Yk)0), the map of biextensions EM → EH, the trivialization of log |EM|, and the height pairing ⟨ , ⟩M : CHm(Yk)0 × CHm′(Yk)0 → R/Q log |k×|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' By (iii) of the lemma in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='3 one has H0 : EH(M) ∼ → EH(H2m+1(Y )(−m)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' For a ∈ CHm(Yk)0, b ∈ CHm′(Yk)0 pick, as in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='3, cycles A, B that represent them such that |A| ∩ |B| = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='3 Let us construct (a, b, κA,B) ∈ EM a,b such that the Hodge realization EH A,B of EM A,B := E(a, b, κA,B) (see 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1) has zero cohomology H0EH A,B equal to the Hodge structure EA,B from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' This would imply that for our M the height pairing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1) equals (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The composition of the maps M(|A|) α→ M(Yk) β→ M(Yk, Yk ∖ |B|) is naturally homotopic to 0: indeed, M(Yk, Yk ∖ |B|) := Cone(M(Yk ∖ |B|) → M(Yk)), and the homotopy κ|A|,|B| is M(|A|) → M(Yk ∖|B|) ⊂ Cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Thus we have (α, β, κ|A|,|B|) ∈ DM(2) (see 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Notice that E(α, β, κ|A|,|B|) = M(Yk ∖ |B|, |A|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' One has Ext−2m(Q(m), M(|A|)) = Zm(|A|) := the group of m-cycles supported on |A| (recall that dim |A| = m), and Ext2m+2(M(Yk, Yk ∖ |B|), Q(m + 1)) = Zm′(|B|) by the Poincar´e duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Therefore we have (αA, Bβ, Bκ|A|,|B|A) = (Q(m)[2m+1], M(Yk), Q(m)[2m+2], αA, Bβ, Bκ|A|,|B|A) ∈ DM(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The promised (a, b, κA,B) ∈ EM a,b is (αA, Bβ, Bκ|A|,|B|A)(−m)[−1 − 2m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The fact that H0EH A,B equals the Hodge structure EA,B from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='3 follows from the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The unipotent nearby cycles in the Hodge setting 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' A nearby cycles reminder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' In this section we play with algebraic varieties over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' For an algebraic variety X we denote by H(X) the abelian category of perverse Hodge Q-sheaves of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Saito on X, by DH(X) its bounded derived category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' It sat- isfies the usual Grothendieck six functors formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Below ∗ is the Verdier duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Every object of H(X), hence of DH(X), carries a canonical weight filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' For F ∈ DH(X) let Γ(X, F), Γc(X, F) ∈ DH be the complex of chains, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' chains with compact support, with coefficients in F equipped with the natural Hodge structure, H· (c)(X, F) := H·Γ(c)(X, F) ∈ H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' set Γ(c)(X) := Γ(c)(X, Q(0)X), H· (c)(X) := H· (c)(X, Q(0)), and denote by ( , ) the Poincar´e duality pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Simi- larly for a closed subvariety A ⊂ X we set ΓA(X) := ΓA(X, Q(0)) ∈ DH, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' 2Indeed, a morphism from a biextension by a trivial group to a trivialized biextension amounts to a biadditive pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' 3Recall that |A|, |B| ⊂ Yk are supports of the cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' HEIGHT PAIRING AND NEARBY CYCLES 9 Let g : X → A1 be a function on X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' set X0 := g−1(0), and let v : X ∖ X0 ֒→ X, iX0 : X0 ֒→ X be the open and closed embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' One has the unipotent nearby cycles functor ψun g : DH(X ∖ X0) → DH(X0) that carries a natural logarithm of monodromy morphism N = Ng = NF : ψun g (F)(1) → ψun g (F) where F ∈ D(X ∖ X0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' It has ´etale local origin with respect to X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' For sheaves on X there is a natural morphism of functors ι : i∗ X0 → ψun g v∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' There are basic canonical identifications: (i) Compatibility with Verdier duality: One has ψun g (F∗) = (ψun g F)∗(1)[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' (ii) Compatibility with proper direct images: Suppose h : X → T is a proper map and t is a function on T such that g = th;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' then one has ψun t h∗F = h∗ψun g F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' (iii) One has Cone(NF) = i∗ X0v∗F(1)[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' (iv) For every n > 0 one has ψun gnF ∼ → ψun g F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' These identifications are mutually compatible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' (i) and (ii) are compatible with the action of N, and (iv) identifies Ngn with nNg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Finally, one has (v) ψun[−1] is t-exact for the perverse t-structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Suppose that X is smooth of dimension n and F = Q(0)X∖X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Then F∗ = F(n)[2n] hence ψun g (F)∗ = (ψun g F)(n − 1)[2n − 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' (a) If g is smooth then ιQ(0)X : Q(0)X0 ∼ → ψun g F, NF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' (b) Suppose g is semi-stable and X0 has two irreducible components Y and Y ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' By (a) one has natural morphisms jY ′∖Y !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='QY ′∖Y → ψun g F → jY ∖Y ′∗QY ∖Y ′ compatible with the N-action (we take it that on the left and right object N acts trivially).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' They form an exact triangle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' its Verdier dual is the same triangle with Y and Y ′ interchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' We are in the setting of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='4 and follow the notation there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Let j : U := X0 ∖ {xα} ֒→ X0 ←֓ {xα} : ⊔ixα be the complementary open and closed embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Let I be the intersection cohomology sheaf j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='∗Q(0)U = τ ≤n−2j∗Q(0)U 4 on X0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' set I+ := π∗Q(0)Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' One has natural self-duality isomor- phisms I∗ = I(n − 1)[2n − 2], I+∗ = I+(n − 1)[2n − 2] (recall that Y is smooth of dimension n − 1 and π is proper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The decomposition theorem for π is easy and explicit: Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' There is a natural orthogonal direct sum decomposition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1) I+ = I ⊕ ⊕αixα∗τ [2,2n−4]Γ(Pα) compatible with the self-dualities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' One has a natural orthogonal direct sum decomposition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2) Γ(Zα) = Hn−2 prim(Zα)[2 − n] ⊕ τ ≤2n−4Γ(Pα) defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Consider the embedding Zα ֒→ Pα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The pullback and Gysin maps Γ(Pα) → Γ(Zα) → Γ(Pα)(1)[2] are mutually dual for the Poincar´e duality pairings, and their composition in either direction equals to the multiplication by c1(O(dα)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='5 Thus the composition of τ ≤2n−4Γ(Pα) → Γ(Zα) → τ ≥0(Γ(Pα)(1)[2]) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' This yields a direct sum decomposition Γ(Zα) =?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='⊕τ ≤2n−4Γ(Pα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' 4Below τ is the usual truncation, pτ is the perverse one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' 5Since O(dα) is the normal bundle to Zα in Pα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' 10 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' BEILINSON Since multiplication by c1(O(dα)) preserves the direct sum decomposition, the only nonzero cohomology of ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' is Hn−2 prim(Zα) ⊂ Hn−2(Zα), q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Consider the embeddings of smooth divisors iZα : Zα ֒→ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' One has i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' ZαQ(0)Y = Q(−1)[−2]Zα, i∗ ZαQ(0)Y = Q(0)Zα, and the composition of the adjunction maps iZα∗i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' ZαQ(0)Y → Q(0)Y → iZα∗i∗ ZαQ(0)Y equals the multiplication by c1(O(−1)) map Q(−1)[−2]Zα → Q(0)Zα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='6 Apply π∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' then i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' xαI+ = Γ(Zα)(−1)[−2], i∗ xαI+ = Γ(Zα) by base change, and the composition of the adjunctions ixα∗i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' xαI+ → I+ → ixα∗i∗ xαI+ is multiplication by c1(O(−1)) map ixα∗Γ(Zα)(−1)[−2] → ixα∗Γ(Zα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Composing the maps τ ≤2n−6Γ(Pα) ֒→ Γ(Zα) and Γ(Zα) ։ τ [2,2n−4]Γ(Pα) that come from decomposition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2) from the left and from the right with the latter ad- junctions, we get the maps ixα∗(τ ≤2n−6Γ(Pα))(−1)[−2] → I+ → ixα∗τ [2,2n−4]Γ(Pα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Their composition is an isomorphism, which yields a decomposition I+ = I?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' ⊕ ixα∗τ [2,2n−4]Γ(Pα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Since the adjunctions are mutually dual, the decomposition is orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2) one has i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' xαI?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' = Hn−2 prim(Zα)(−1)[−n] ⊕ Q(n − 1)[2 − 2n], i∗ xαI?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' = Hn−2 prim(Zα)[2 − n] ⊕ Q(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Thus I?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' [n − 1] is a perverse sheaf which equals Q(0)[n − 1]U on U and has no subquotients supported on {xα}, and so I?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' We are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' □ Remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' (i) The adjunction map Q(0)X0 → π∗Q(0)Y = I+ takes value in I ⊂ I+ since Hom(Q(0)X0, ixα∗τ [2,2n−4]Γ(Pα)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' (ii) Set B := ⊕ixα∗Hn−2 prim(Zα)[1 − n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' By the formula for i∗ xαI at the end of the previous paragraph, one has an exact triangle Q(0)X0 → I → B[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' As in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='5, t is a local coordinate at 0 ∈ S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' shrinking S we can assume that t is defined and invertible on S ∖ {0}, so X0 = (tf)−1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Consider the functor ψun tf : DH(X ∖ X0) → DH(X0) (see 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Set R := ψun tf Q(0)X∖X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' By 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1(i) one has a canonical self-duality identification R�� = R(n − 1)[2n − 2] and the mutually dual maps Q(0)X0 ι→ R ι∗ → Q(0)∗ X0(1 − n)[2 − 2n] which are isomorphisms over U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The next result is due to Illusie [Il];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' we will need it in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The reader can skip it at the moment and jump directly to section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' For every critical point xα one has canonical isomorphisms (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1) i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' xαR = Γc(Pα ∖ Zα), i∗ xαR = Γ(Pα ∖ Zα) interchanged by the duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The N-action on i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' xαR, i∗ xαR is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' (a) The claim is local at xα, so for the proof we remove from X the rest of critical points, and still call it X by the abuse of notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Let S♭ → S be the covering of degree dα obtained by adding t♭ = t1/dα to the sheaf of functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' its Galois group is µdα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Set X♭ := X×SS♭ and let f ♭ : X♭ → S♭ be the projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Our X♭ is a hypersurface {(x, t♭) : (tf)(x) − t♭dα = 0} in X × A1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' its only singular point is (xα, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The projectivized tangent cone Qα of X♭ at (xα, 0) is a hypersurface in P + α := P(T(xα,0)X × A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The Galois group µdα acts on X♭ hence on Qα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' (b) Let us check that Qα is a µdα-covering of Pα completely ramified along Zα and ´etale over its complement, and Qα is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' To see this, consider the leading term [tf]dα(x) (of the Taylor expansion) of tf at xα;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' then the leading term of 6Since O(−1) is the normal bundle to Zα in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' HEIGHT PAIRING AND NEARBY CYCLES 11 (tf)(x) − t♭dα at (xα, 0) is [tf]dα(x) − t♭dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The zeros of [tf]dα is Zα ⊂ Pα, of [tf]dα(x) − t♭dα is Qα ⊂ P + α , and so the projection Qα → Pα (x, t♭) �→ x, is as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The smoothness of Qα follows from that of Zα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' (c) Let π+ : X+ → X♭ be the blowup of X♭ at (xα, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' By (b) X+ is smooth and the map f + := f ♭π+ : X+ → S♭ has semistable reduction at 0 ∈ S♭.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The fiber X+ 0 has two irreducible components: one equals Y and the other Qα, and their intersection equals Zα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The action of µdα on X♭ yields one on X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The µdα-action on X+ 0 fixes Y and acts on Qα as described in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The projection π+ 0 : X+ 0 → X♭ 0 = X0 contracts Qα to xα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Set R+ := ψun tf +Q(0)X+∖X+ 0 , R♭ := ψun tf ♭Q(0)X♭∖X♭ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' These are sheaves on X+ 0 and X♭ 0 = X0 respectively that are naturally µdα-equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' By 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1(ii) (with h = π+) one has a natural identification π+ 0∗R+ = R♭ compatible with the µdα- actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Since the projection p : X♭ → X is a µdα-torsor over X ∖ X0 one has Q(0)X∖X0 = (p∗Q(0)X♭∖X♭ 0)µdα , and so, by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1(ii) with h = p, one has R = R♭µdα .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Therefore R = (π+ 0∗R+)µdα .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' (d) By 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1(iv) with g = t♭f +, n = dα, one has ψun tf + = ψun t♭f +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Our t♭f + is semi- stable, so we have the exact triangle jY ∖Zα!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='QY ∖Zα → R+ → jQα∖Zα∗QQα∖Zα as in Example (b) in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Applying π+ 0∗ we get an exact triangle j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='QU → R♭ → ixα∗Γ(Qα∖Zα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Passing to µdα-invariants we get, by (b), an exact triangle j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='QU → R → ixα∗Γ(Pα ∖ Zα);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' here we use the identification Γ(Qα ∖ Zα)µdα ∼ → Γ(Pα ∖ Zα) defined as the composition Γ(Qα ∖ Zα)µdα ⊂ Γ(Qα ∖ Zα) tr → Γ(Pα ∖ Zα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Thus we get the isomorphism i∗ xαR ∼ → Γ(Pα ∖ Zα) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The second isomorphism there comes in the dual manner from the dual exact triangle jQα∖Zα!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='QQα∖Zα → R+ → jY ∖Zα∗QY ∖Zα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Since π+ 0∗ commutes with duality, the two isomorphisms are mutually dual, and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' □ Let αR be the composition B ∂→ Q(0)X0 ι→ R where ∂ is the boundary map of the triangle from Remark (ii) in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2, so I = Cone(∂).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Let us compute the map i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' xα(αR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Consider the standard triangle Hn−2 prim(Zα)[1−n] δ→ Γc(Pα ∖Zα) tr → Q(1−n)[2−2n] that comes from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' −i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' xα(αR) equals the composition δR of the maps Hn−2 prim(Zα)[1 − n] δ→ Γc(Pα ∖ Zα) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1) = i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' xαR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Consider the exact triangle (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2) jQα∖Zα!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='Q(0)Qα∖Zα ⊕ jY ∖Zα!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='Q(0)Y ∖Zα → Q(0)X+ 0 → Q(0)Zα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Let (δQ, δY ) : Q(0)Zα[−1] → jQα∖Zα!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='Q(0)Qα∖Zα ⊕ jY ∖Zα!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='Q(0)Y ∖Zα be the bound- ary map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Its composition with the map to Q(0)X+ 0 , and hence with the further composition with Q(0)X+ 0 ι→ R+, is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Therefore the sum of the compositions Q(0)Zα[−1] δQ −→ jQα∖Zα!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' ι��� R+ and Q(0)Zα[−1] δY −→ jY ∖Zα!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' ι→ R+ is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Ap- ply i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' xαπ+ ∗ and consider the restriction of our compositions to Hn−2 prim(Zα)[1 − n] ⊂ Γ(Zα)[−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' For the first one it is δR, for the second one it is i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' xα(αR), and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' □ 12 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' BEILINSON 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Set P := R[n − 1] = ψun tf Q(0)X∖X0[n − 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' this is a perverse sheaf on X0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' one has a canonical self-duality identification P∗ = P(n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Consider the perverse sheaves PN := Ker(N : P → P(−1)), PN := Coker(N : P(1) → P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' (i) Q(0)X0[n − 1] is a perverse sheaf of weights n − 1 and n − 2 with grW n−1 = I[n − 1], grW n−2 = ⊕α ixα∗Hn−2 prim(Zα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' (ii) One has PN = Q(0)X0[n − 1], PN = (Q(0)X0[n − 1])∗(1 − n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' (iii) P has weights in [n − 2, n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' One has Wn−1P = Q(0)X0[n − 1], P/Wn−2P = (Q(0)X0[n − 1])∗(1 − n), grW n−2P = ⊕α ixα∗Hn−2 prim(Zα), grW n−1P = I[n − 1], grW n P = (grW n−2P)∗(1 − n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' (i) The exact triangle from Remark (ii) in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2 amounts to an exact triangle ⊕ixα∗Hn−2 prim(Zα) → Q(0)X0[n − 1] → I[n − 1], and we are done since its left and right terms are pure perverse sheaves of weights n − 2 and n − 1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' (ii) For any sheaf A on X one has a canonical exact triangle i∗ X0A → i∗ X0v∗v∗A → i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' X0A[1]: Indeed, the map v!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='v∗A → v∗v∗A factors as composition v!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='v∗A → A → v∗v∗A, and so one has an exact triangle Cone(v!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='v∗A → A) → Cone(v!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='v∗A → v∗v∗A) → Cone(A → v∗v∗A) which is supported on X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The promised exact triangle is its restriction to X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Now take for A the perverse sheaf Q(0)X[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The first term of the triangle is Q(0)X0[n] which is perverse sheaf shifted by 1, its third term is (Q(0)X0[n−1])∗(−n) which is a perverse sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Therefore they equal, respectively, pH−1 and pH0 of i∗ X0v∗v∗Q(0)X[2n], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=', of Cone(N : P → P(−1)) by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1(iii), and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' (iii) Since N is nilpotent, the weights of P are bounded from below by the minimum of weights of PN, which is n − 2 by (ii) and (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' By self-duality of P they are bounded then from above by n, and we have the first assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' It implies that Wn−2P ⊂ PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The rest follows directly from (i), (ii), and self-duality of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Proof of the proposition in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' We use the notation in loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Injectivity of sp : (ψun t H)N → Hn−1(X0) follows from the local invariant cycles theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Let us check the surjectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' By 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1(ii) applied to h = f (recall that f is proper) and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1(v) applied to ψun t , one has ψun t H = H0(X0, P)(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' By 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='4 we have exact sequence of perverse sheaves 0 → ⊕α ixα∗Hn−2 prim(Zα)(n−1) → P(n−1) → (Q(0)X0[n−1])∗ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Its left term has finite support, and so has no cohomology in degrees ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Therefore the map H0(X0, P)(n − 1) → H0(X0, (Q(0)X0[n − 1])∗) = Hn−1(X0) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' This map equals sp, and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' □ §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The motivic setting and the construction of EψM a,b ∈ EM a,b 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' We are in the setting of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='8 so k ⊂ C is a subfield and we play with varieties over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Changing slightly the notation of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='3 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='8, for a variety Z = Zk we set ZC := Z ⊗k C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The notation of §3 is preserved except that we equip from now on all Hodge sheaves and Hodge structures met previously with extra upper index H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' We play with motives (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' motivic sheaves) over varieties, see [A1] and [CD].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' For a variety Z the category of constructible Q-motives over Z is denoted by DM(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' We use Grothendieck’s six functors formalism for DM as developed in [CD].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Recall that DM(Spec k) = DM(k) is the category of Voevodsky’s geo- metric Q-motives over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' For a variety Z one has M(Z) = πZ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' ZQ(0) where πZ : Z → Spec k is the structure map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' For a motivic sheaf F on Z set Γ(Z, F) := πZ∗F, Γc(Z, F) := πZ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='F ∈ DM(k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' we write Γ(c)(Z) := Γ(c)(Z, Q(0)Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' There is HEIGHT PAIRING AND NEARBY CYCLES 13 a Hodge realization functor DM(Z) → DH(ZC), F �→ FH, compatible with the six functors and the Verdier duality ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' For a smooth Z of dimension d one has π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' ZQ(0) = Q(d)Z[2d], and so M(Z) = Γc(Z)(d)[2d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The formalism of unipoteny nearby cycles in the setting of motivic sheaves was developed in §§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='4, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='6 of [A2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The motivic version of everything said in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1 holds except property (v) (for the t-structure is not available).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The Hodge realization functor commutes with the nearby cycles functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Notation: Notice that Hom(Q(i)[2i], Q(j)[2j]) is 0 if i ̸= j and Q for i = j,7 and so every object M ∈ M(k) which is isomorphic to a direct sum of motives Q(i)[2i], i ∈ Z, can be written in a unique manner as ⊕i Vi(i)[2i] where Vi is a vector space (then Vi = Hom(Q(i)[2i], M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Set τ ≤2aM := ⊕i≥−a Vi(i)[2i], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' We are in the situation of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2 in the setting of k-varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' As in loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=', I+ := π∗Q(0)Y ∈ DM(X0) (so I+H is the corresponding Hodge sheaf from loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=') Since Y is smooth and π is proper one has a natural self-duality I+∗ = I+(n−1)[2n−2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The t-structure in DM is not available, so we define the motivic intersection cohomology sheaf I using a motivic version of decomposition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1): Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' There is a natural orthogonal direct sum decomposition in DM(X0) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1) I+ = I ⊕ ⊕αixα∗τ [2,2n−4]Γ(Pα) whose Hodge realization is (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' It repeats the proof in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2 (minus its last paragraph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Namely, we first define a natural orthogonal decomposition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2) Γ(Zα) = Hn−2 prim(Zα)[2 − n] ⊕ τ ≤2n−4Γ(Pα) in DM(xα) = DM(kxα) whose Hodge realization is (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='8 The construction in loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' uses only basic six functors functoriality, so we can repeat it literally in the motivic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Then we proceed to define (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1) as in loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' □ Set B := ⊕α ixα∗Hn−2 prim(Zα)[1 − n] ∈ DM(X0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The self-dualities of Γ(Zα) and of I+, and the above orthogonal decompositions yield natural self-dualities (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='3) B∗ ∼ → B(n − 2)[2n − 2], I∗ ∼ → I(n − 1)[2n − 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' (i) The adjunction χ : Q(0)X0 → π∗Q(0)Y = I+ takes values in I ⊂ I+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' (ii) One has Cone(χ : Q(0)X0 → I) = B[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' (i) Follows since Hom(Q(0)X0, ixα∗τ [2,2n−4]Γ(Pα)) = Hom(Q(0), τ [2,2n−4]Γ(Pα)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' (ii) Since χ|U = idQ(0)U the cone Cone(χ) is supported on {xα}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Now i∗ xαCone(χ) = 7This follows since M(Pn) = ⊕i∈[0,n]Q(i)[2i] and End(M(Pn)) = CHn(Pn × Pn) = Q[0,n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' 8So Hn−2 prim(Zα) is a notation for a motive whose Hodge realization is the primitive cohomology of Zα;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' its definition does not involve any cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' To construct it explicitly, choose a k-point z in Pα ∖ Zα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Let πz : Zα → Pn−2 be the corresponding projection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' this is a finite map of degree dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Then Hn−2 prim(Zα) is the kernel of the projector d−1 α πt zπz acting on M(Zα)(2 − n)[4 − 2n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' 14 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' BEILINSON Cone(i∗ xα(χ)) equals Hn−2 prim(Zα)[2 − n] by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2) and the construction of I, q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' □ Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Since Exti(Q(0)X0, Q(0)∗ X0(1−n)[2−2n]) = Exti(Q(0), M(X0)(1−n)[2− 2n]) = CHn−1(X0, −i) we see that Ext0 = Zn−1(X0) and Ext̸=0 = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=', one has Hom(Q(0)X0, Q(0)∗ X0(1 − n)[2 − 2n]) = Zn−1(X0) = Zn−1(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' One has χ∗χ = ǫ where ǫ : Q(0)X0 → Q(0)∗ X0(1 − n)[2 − 2n] is the map that corresponds to the sum of irreducible components cycle (it is enough to check the assertion on U where it is obvious).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' We are in the situation of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='3 in the setting of k-varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Consider the functor ψun tf : DM(X ∖ X0) → DM(X0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' There is a canonical morphism ι : i∗ X0 → ψun tf v∗ of functors on DM(X) and its Verdier dual ι∗ : ψun tf v∗ → i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Therefore we have a motivic sheaf R := ψun tf Q(0)X∖X0 equipped with a natural self-duality R∗ ∼ → R(n − 1)[2n − 2] and mutually dual maps Q(0)X0 ι→ R ι∗ → Q(0)∗ X0(1 − n)[2 − 2n] that are isomorphisms over U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Let ∂ : B → Q(0)X0 be the boundary map of the triangle from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='3(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Set αR := ι∂ : B → R, and let βR be α∗ R combined with the self-duality identifications for R and B, so we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1) B αR −→ R βR −→ B(−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Lemma-construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The composition βRαR is homotopic to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' In fact, there is a canonical up to a homotopy κR such that d(κR) = βRαR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' By Remark and Example in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='3 one has βRαR = ∂∗ι∗ι∂ = ∂∗ǫ∂ = ∂∗χ∗χ∂ = (χ∂)∗χ∂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Notice that χ∂ is homotopic to 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' choose a homotopy λ, d(λ) = χ∂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Now set κR := λ∗χ∂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Independence of κR up to a homotopy from the choice of λ: if λ′ is another homotopy as above, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=', d(λ) = d(λ′), then κ′ R = λ′∗χ∂ = κR + (λ′ − λ)χ∂ = κR + d((λ − λ′)λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' □ Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Our κR is self-dual up to homotopy: Indeed, one has κ∗ R = (χ∂)∗λ = κR + d(λ∗λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Below we use the notation from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' We have defined (αR, βR, κR) ∈ DM(X0)(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' It yields the objects ER := E(αR, βR, κR) ∈ DM(X0) and (αI, βI, κI) := σ(αR, βR, κR) ∈ DM(X0)(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' As follows from Remark in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='4 and the defini- tions, the above three objects are naturally self-dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' There is a homotopy equivalence θ : I ∼ → ER such that the maps βIθ : I → B[1], θ−1αI : B(−1)[−1] are a morphism of the triangle in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='3(ii) and its dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Our θ is unitary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=', θ∗ = θ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Recall that we have a natural homotopy equivalence (λ, χ) : Cone(∂ : B → Q(0)X0) ∼ → I (see 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='3(ii)), and ER is the direct sum B[1] ⊕ R ⊕ B(−1)[−1] with (αR, −κR, βR) added to the differential (see 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Our θ is the composition I ∼ ← Cone(∂) θ′ → ER where θ′ is the next morphism: its restriction to B[1] ⊂ Cone(∂) identifies it with the first summand in ER, and its restriction to Q(0)X0 ⊂ Cone(∂) is (0, ι, −λ∗χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' HEIGHT PAIRING AND NEARBY CYCLES 15 One has θ∗θ = idI: we need to check that θ′∗ρθ′ = (λ, χ)∗(λ, χ) : Cone(∂) → Cone(∂∗) where ρ : ER ∼ → E∗ R(1 − n)[2 − 2n] is the self-duality for ER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' As follows from Remark in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='4, ρ is the matrix with the self-dualities for R and B’s on the diagonal and the only non-zero off-diagonal entry being λ∗λ : B → B∗(1−n)[2−2n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The rest is an immediate calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The assertion that βIθ is the morphism of the triangle in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='3(ii) means that βIθ′ is the projection Cone(∂) → B[1] which is evident from the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The assertion that αIθ is dual to βIθ follows from the unitarity of θ once we know that θ is a homotopy equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Let us check it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Our θ′ is a morphism Cone(B → Q(0)X0) → Cone(B → Cone(βR)[−1]) com- patible with the projections to B, and so it is enough to check that the map (ι, −λ∗χ) : Q(0)X0 → Cone(βR)[−1] is a homotopy equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Since ι is a homo- topy equivalence on U, it is enough to check our claim after applying i∗ xα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' The story of section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='3 uses only the six functors formalism and basic facts from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1, so it remains literally true in the motivic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Consider the canonical homotopy equivalence a : i∗ xαR ∼ → Γ(Pα ∖ Zα) of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' By the Verdier dual assertion to the lemma in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='3, a identifies i∗ xα(βR) with minus the residue map r : Γ(Pα ∖ Zα) → Hn−2 prim(Zα)(−1)[1 − n] ⊂ Γ(Zα)(−1)[−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='2) we have a split exact triangle Q(0) → Γ(Pα ∖ Zα) r→ Hn−2 prim(Zα)(−1)[1 − n], so a identifies i∗ xαCone(βR)[−1]) with Q(0) ⊂ Γ(Pα∖Zα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' It follows directly from the construction of a that ai∗ xα(ι) coincides with the latter embedding, and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Proof of the theorem in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' We have (αI, βI, κI) ∈ DM(X0)(2), hence Γ(αI, βI, κI) ∈ DM(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' For two Bloch cycles A, B of classes clA, clB ∈ Hom(Q(0), Hn−2 prim(Zα)(m)) we have (cl∗ A, clB∗)Γ(αI, βI, κI) ∈ EM(Γ(I)(m − 1)[1 − n]) = EM(Γ(I+)(m − 1)[1 − n]) = EM(M) where M := M(Y )(−m)[−1 − 2m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' By the construction the Hodge realization embedding EM(M) ֒→ EH(M) = EH(Hm(Y )) identifies it with Eψ A,B from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content='6, and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' □ References [A1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' Ayoub, Les six op´erations de Grothendieck et le formalisme des cycles ´evanescents dans le monde motivique (I), Ast´erisque 314, SMF, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E0T4oBgHgl3EQfwQGR/content/2301.02630v1.pdf'} +page_content=' [A2] J.' metadata={'source': 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