derived categories of twisted sheaves on calabi-yau manifolds

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$Lecture notes for Math 522 Spring 2012 (Rudin chapter 7)$

$Math 320 - Fall 2010 HW #7 Selected Solutions Problem 5.1.32 Let ...$

- if P = Q, Ext i X(F 0 (P ), F 0 (Q)) = 0, for all i; - Ext i X(F 0 (P ), F 0 (P )) = 0, for all P , and for i > 3; - HomX(F 0 (P ), F 0 (P )) = C, for all P ; - F 0 (P ) ⊗ ωX ∼ = F 0 (P ), for all P . 104 The last property follows easily from the adjunction formula and the fact that the fibers <strong>of</strong> X have trivial canonical bundles. If P and Q lie in different fibers <strong>of</strong> ¯ J → S, there is nothing to prove as Supp F 0 (P ) and Supp F 0 (Q) are disjoint. So assume that P and Q lie in the same fiber ¯ Js (over some s ∈ S). Let i : Xs → X be the inclusion, so that F 0 (P ) = i∗F (P ) and F 0 (Q) = i∗F (Q). Using Proposition 3.3.5 we reduce to checking that - if P = Q, Ext i Xs (F (P ), F (Q)) = 0 for all i; - Ext i Xs (F (P ), F (P )) = 0 for all P , and for i > 1; - HomXs(F (P ), F (P )) = C for all P . For irreducible fibers (smooth, nodal or cuspidal), the <strong>sheaves</strong> F (P ) are stable, so the first and third statements follow from [25, 1.2.7 and 1.2.8], while for I2 fibers they follow from Lemma 6.3.6. Since Xs is Cohen-Macaulay and has trivial dualizing sheaf, we get by Serre duality Ext i Xs (F (P ), F (Q)) = Ext1−i(F (Q), F (P )). Xs which proves the second statement, thus finishing the pro<strong>of</strong>. Since the actual result is quite central to the theory, we restate it in a slightly different form: Theorem 6.5.4. Let X → S be a generic elliptic Calabi-Yau threefold, let J → S be its relative Jacobian, and let ¯ J be an analytic small resolution <strong>of</strong> the singularities <strong>of</strong> J. Let U = S \∆, where ∆ is the discriminant locus <strong>of</strong> X → S. If α ∈ Br(JU) is the obstruction to the existence <strong>of</strong> a universal sheaf on XU ×U JU (or, alternatively, the element <strong>of</strong> the Tate-Shafarevich group representing X → S), then there is a unique extension <strong>of</strong> α to an element ¯α <strong>of</strong> Br( ¯ J). For this ¯α, there is an equivalence <strong>of</strong> <strong>derived</strong> <strong>categories</strong> D b coh(X) ∼ = D b coh( ¯ J, ¯α −1 ). Remark 6.5.5. The fact that X is a Calabi-Yau threefold is irrelevant for the actual pro<strong>of</strong>. We only needed that KX is trivial along the fibers <strong>of</strong> X → S, which is an immediate consequence <strong>of</strong> the adjunction formula, if X → S has fibers which are plane curves <strong>of</strong> arithmetic genus 1. As an easy corollary <strong>of</strong> the above theorem, we obtain the following result:
Corollary 6.5.6. Under the hypotheses <strong>of</strong> the previous theorem, ¯ J has trivial canonical class. 105 Pro<strong>of</strong>. The result follows at once when one notes that the Serre functors in D b coh (X) and D b coh ( ¯ J, ¯α −1 ) must be the same (being defined categorically), and these determine the respective canonical classes. 6.6 Equivalent Twistings The result we want to focus on in this section is the following: Theorem 6.6.1. Let X → S be a generic elliptic Calabi-Yau threefold, and let J → S be its relative Jacobian. Let ¯ J → J be any analytic small resolution <strong>of</strong> the singularities <strong>of</strong> J, let ¯α ∈ Br( ¯ J) be the twisting defined in Theorem 6.5.4, and let n be the order <strong>of</strong> ¯α in Br( ¯ J). Then we have for any k coprime to n. D b coh( ¯ J, ¯α) ∼ = D b coh( ¯ J, ¯α k ), This result should be compared to the similar one for K3 surfaces (Theorem 5.5.2), and contrasted to the corresponding situation for spectra <strong>of</strong> local rings (Theorem 1.3.19). Theorem 6.6.2. Let X → S be a generic elliptic Calabi-Yau threefold, and let M → S be the relative moduli space <strong>of</strong> semistable <strong>sheaves</strong> <strong>of</strong> rank 1, degree k on the fibers <strong>of</strong> X → S, for an appropriate choice <strong>of</strong> relatively ample sheaf on X → S. Let Y be the union <strong>of</strong> the components <strong>of</strong> M that contain a point corresponding to a stable line bundle on a fiber <strong>of</strong> X → S. Then Y is in fact smooth, irreducible, a universal sheaf exists on X ×S Y , and D b coh(X) ∼ = D b coh(Y ). Notation 6.6.3. In keeping with the notation introduced in Section 4.5, we will denote Y by X k . Before we proceed to the pro<strong>of</strong> <strong>of</strong> this theorem, let us remark that Theorem 6.6.1 is an immediate consequence <strong>of</strong> this result: indeed, it is easy to see that Y is again a generic elliptic Calabi-Yau threefold: the equivalence <strong>of</strong> <strong>categories</strong> D b coh (X) ∼ = D b coh (Y ) implies KY = 0 by the uniqueness <strong>of</strong> the Serre functor. Also, the isometry induced by the Fourier-Mukai transform (Corollary 3.1.13 and Proposition 3.1.14), induces an isomorphism H 2,0 (X) ⊕ H 4,0 (X) ∼ = H 2,0 (Y ) ⊕ H 4,0 (Y ), and therefore H 2,0 (Y ) = 0. By Serre duality, this implies H 1 (Y, OY ) = 0, so that Y is Calabi-Yau. Since X and Y are locally isomorphic over S, they obviously have
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DERIVED CATEGORIES OF TWISTED SHEAV
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DERIVED CATEGORIES OF TWISTED SHEAV
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Tuturor celor din care am fost făc
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Table of Contents
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List of Figures 6.
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Introduction and Overview Twisted <
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twisted by α ∈
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fibers introduces interesting featu
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Chapter 1 Twisted Sheaves In this c
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to be H2 an(X, O∗ X ). If we do n
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is bijective. Then A is called an A
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such that α = δa and β = δb. Si
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Lemma 1.2.3. Let α ∈ Č2 (X, O
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Proof. Let U ′
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Proof. Since E and
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Proof. Follows fro
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Definition 1.3.12. Let A be a ring.
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Definition 1.3.18. Two R-algebras A
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property “free on stalks” for t
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2.2 The Derived Category and Derive
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to reduce to the case when F · als
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So assume n > 0, and as before cons
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Proposition 2.3.4. Let f : X → Y
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2.4 Duality for Proper Smooth Morph
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complex manifolds, and let X × Y
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if v consists only of</stro
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The reason for the notation is that
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Proof. The first s
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which by definition associates to a
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Part II Applications 49
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X which would intersect a general f
Page 63 and 64: to C ′ , independent of</
Page 65 and 66: Jij → S. This means that if Fij w
Page 67 and 68: Li = ρ∗ i L −1 |Xi . Then we h
Page 69 and 70: It is easy to see that ϕi(ti) = [O
Page 71 and 72: Remark 4.5.3. Note that we have fix
Page 73 and 74: allow us to get X; however, finding
Page 75 and 76: endowed with the product ((r, l, s)
Page 77 and 78: Theorem 5.1.10. Under the hypothese
Page 79 and 80: groups of differen
Page 81 and 82: Let c be the image of</stro
Page 83 and 84: Proof. Trivial cha
Page 85 and 86: and therefore rk(V ) = (v(V ), (0,
Page 87 and 88: for some k ≫ 0. By the same argum
Page 89 and 90: and therefore Now consider the map
Page 91 and 92: In order to identify the kernel, no
Page 93 and 94: for all F , G stable sheave
Page 95 and 96: Chapter 6 Elliptic Calabi-Yau Three
Page 97 and 98: Definition 6.1.2. A projective morp
Page 99 and 100: Example 6.2.1. We’ve already seen
Page 101 and 102: contract 45 curves 2H+D-E contract
Page 103 and 104: Remark 6.3.2. This theorem basicall
Page 105 and 106: But this is impossible, because the
Page 107 and 108: We have H 0 (C, IQ) = 0 and χ(IQ)
Page 109 and 110: Having fixed a smooth point P ∈ C
Page 111 and 112: 101 Proof. Since X
Page 113: 103 Therefore, α can be represente
Page 117 and 118: 107 Remark 6.6.4. We could also pro
Page 119 and 120: 109 where the intersection form on
Page 121 and 122: 111 where ci = ci(i∗U · ). Obvio
Page 123 and 124: 113 One space of i
Page 125 and 126: Open Questions and Further Directio
Page 127 and 128: Bibliography [1] Aspinwall, P., Mor
Page 129 and 130: 119 [28] D. Bayer and M. Stillman,
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derived categories of twisted sheaves on calabi-yau manifolds

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