derived categories of twisted sheaves on calabi-yau manifolds

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

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

under the identification H 2 (X, Z) ∼ = H 2 (X0, Z) and the inclusion (H 2 (X, Z)/NS(X)) ⊗ Q/Z ↩→ Br ′ (X). The interpretation <strong>of</strong> this theorem is the following: if we try to deform a vector bundle E0, given on the central fiber X0, in a family in which the class c1(E0) is not algebraic in neighboring fibers Xt, the only hope to be able to do this is to deform E0 as a <strong>twisted</strong> sheaf, and then the twisting should be precisely − 1 rk(E0) c1(E0). We first need a couple <strong>of</strong> propositions: Lemma 5.2.2 (The Roman 9 Lemma). Let A be an abelian category with enough injectives, F a left-exact functor, and let H i be the right <strong>derived</strong> functors <strong>of</strong> F . Consider a commutative diagram 0 0 ❩ ❩❩⑦ ✚ ✚ ✚❃ 0 ✲ A ✲ B ❩ ❩❩⑦ ✲ C ✚ ✚ ✲ 0 ✚❃ ✚ D ✚✚❃ ❩ 0 ✲ A ✲ E ❩ ❩⑦ ✲ F ✲ 0 ✚ ✚✚❃ ❩ 0 ❩ ❩⑦ 0 with exact rows and diagonals (A, B, C, D, E, F ∈ Ob(A )). Then the induced diagram in cohomology H i (A) ✲ H i (B) ✲ H i (C) ✲ H i+1 (A) ❩ ❩❩⑦ ✚ ✚ ✚❃ H i (D) ✚ ✚✚❃ ❩ ❩ ❩⑦ H i (A) ✲ i H (E) ✲ i H (F ) ✲ i+1 H (A) commutes as well, except for the right pentagon which anti-commutes. Pro<strong>of</strong>. The only non-trivial issue is the anti-commutativity <strong>of</strong> the right pentagon; we’ll prove it only for the case when A is the category <strong>of</strong> quasi-coherent <strong>sheaves</strong> on a scheme X, and one uses Čech cohomology. In the general case <strong>of</strong> an abelian category, this is an easy exercise in homological algebra (done for example in [44]). Let d be an element <strong>of</strong> Hi (D), and represent it as a Čech cocycle {dj0...ji } over some fine enough cover <strong>of</strong> X. From here on we’ll omit the indices, and just refer to this collection as d. 70
Let c be the image <strong>of</strong> d under the natural map D → C (as Čech cocycles). By possibly refining the cover, one can lift c to a cochain b ∈ Či (B). It is the failure <strong>of</strong> b to be a cocycle that gives the coboundary map H i (C) → H i+1 (A). Let d ′ be the cochain obtained by applying to b the map B → D. By the commutativity <strong>of</strong> the triangle that contains B, C and D, d ′ maps to c as well, so that d − d ′ maps to 0 in c, so d − d ′ lifts to a cochain in E. Let this cochain be e. Now e is a lift <strong>of</strong> f (where f is the image <strong>of</strong> d under the map D → F ): d ′ comes from b, so it maps to 0 in F . Computing, we have ∂e = ∂(d − d ′ ) = −∂d ′ = −∂b, which is what we needed. Definition 5.2.3. Consider the two exact sequences on a space X and 0 → O ∗ X → GL(n) → PGL(n) → 0 0 → Z/nZ → SL(n) → PGL(n) → 0, which we’ve already seen in Section 1.1 (as usual, we are using the étale or analytic topologies). For an element p <strong>of</strong> H 1 (X, PGL(n)), let a(p) and t(p) be the images <strong>of</strong> p under the two coboundary maps and H 1 (X, PGL(n)) → H 2 (X, O ∗ X) H 1 (X, PGL(n)) → H 2 (X, Z/nZ), respectively. We’ll call a(p) the analytic twisting class <strong>of</strong> p and t(p) the topological twisting class <strong>of</strong> p. The first one belongs to Br ′ (X), and as such depends on the complex structure <strong>of</strong> X, while the second one is in H 2 (X, Z/nZ) and depends only on the topology <strong>of</strong> X. If Y → X is a P n−1 -bundle over X, then the analytic and topological twisting classes <strong>of</strong> Y/X, t(Y/X) and a(Y/X), are the classes <strong>of</strong> the element p <strong>of</strong> H 1 (X, PGL(n)) associated to the bundle Y → X. Proposition 5.2.4. Let X be a scheme, E a rank n locally free sheaf on X and let Y = Proj(E ) → X be the associated projective bundle. Then t1(Y/X) = −c1(E ) mod n. (Here, and in the sequel, by reducing mod n we mean applying the natural map H 2 (X, Z) → H 2 (X, Z/nZ)). 71
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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
Page 25 and 26:
Lemma 1.2.3. Let α ∈ Č2 (X, O
Page 27 and 28:
Proof. Let U ′
Page 29 and 30: Proof. Since E and
Page 31 and 32: Proof. Follows fro
Page 33 and 34: Definition 1.3.12. Let A be a ring.
Page 35 and 36: Definition 1.3.18. Two R-algebras A
Page 37 and 38: property “free on stalks” for t
Page 39 and 40: 2.2 The Derived Category and Derive
Page 41 and 42: to reduce to the case when F · als
Page 43 and 44: So assume n > 0, and as before cons
Page 45 and 46: Proposition 2.3.4. Let f : X → Y
Page 47 and 48: 2.4 Duality for Proper Smooth Morph
Page 49 and 50: complex manifolds, and let X × Y
Page 51 and 52: if v consists only of</stro
Page 53 and 54: The reason for the notation is that
Page 55 and 56: Proof. The first s
Page 57 and 58: which by definition associates to a
Page 59 and 60: Part II Applications 49
Page 61 and 62: 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: groups of differen
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 and 114: 103 Therefore, α can be represente
Page 115 and 116: Corollary 6.5.6. Under the hypothes
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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