derived categories of twisted sheaves on calabi-yau manifolds

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

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

Part I Theoretical Results 6
Chapter 1 Twisted Sheaves In this chapter we introduce <strong>twisted</strong> <strong>sheaves</strong>, which are the main object <strong>of</strong> study in this work. We begin with the definition <strong>of</strong> the Brauer group through various descriptions <strong>of</strong> its elements (as Čech 2-cocycles, as Azumaya algebras, as gerbes), and we continue with the definition and basic properties <strong>of</strong> <strong>twisted</strong> <strong>sheaves</strong>. These are viewed as local <strong>sheaves</strong> and isomorphisms that don’t quite “match up” in Section 1.2. Section 1.3 gives a more natural interpretation <strong>of</strong> <strong>twisted</strong> <strong>sheaves</strong>, as modules over an Azumaya algebra. The material in this chapter is fundamental in the understanding <strong>of</strong> the sequel. 1.1 H2 ét (X, O∗ X ), the Brauer Group and Gerbes The Cohomological Brauer Group Let us start with an example, which will serve as the main motivation for the introduction <strong>of</strong> the Brauer group. Example 1.1.1. Let f : Y → X be a smooth morphism <strong>of</strong> smooth algebraic varieties or complex analytic spaces, such that all the fibers <strong>of</strong> f are isomorphic to P n−1 (this will be called a projective bundle). We are interested in answering the question: is there a vector bundle E on X, <strong>of</strong> rank n, such that Y → X is isomorphic to P(E ) → X? A possible way to study this problem is the following: first, solve the problem locally on X, using a fine enough topology (étale, or analytic). We can do this because locally in these topologies any projective bundle is trivial, so projectivizing the trivial bundle <strong>of</strong> rank n gives a solution. Second, cover X with open sets Ui such that over Ui we can solve the problem, let Yi = f −1 (Ui), and on each Ui choose a vector bundle Ei and an isomorphism ϕi : P(Ei) ∼ −→ Yi over Ui. Let ϕij = ϕ −1 i ◦ ϕj : P(Ej|Ui∩Uj ) → P(Ei|Ui∩Uj ). 7
Page 1 and 2: DERIVED CATEGORIES OF TWISTED SHEAV
Page 3 and 4: DERIVED CATEGORIES OF TWISTED SHEAV
Page 5 and 6: Tuturor celor din care am fost făc
Page 7 and 8: Table of Contents
Page 9 and 10: List of Figures 6.
Page 11 and 12: Introduction and Overview Twisted <
Page 13 and 14: twisted by α ∈
Page 15: fibers introduces interesting featu
Page 19 and 20: to be H2 an(X, O∗ X ). If we do n
Page 21 and 22: is bijective. Then A is called an A
Page 23 and 24: 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
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Remark 4.5.3. Note that we have fix
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allow us to get X; however, finding
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endowed with the product ((r, l, s)
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Theorem 5.1.10. Under the hypothese
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groups of differen
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Let c be the image of</stro
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Proof. Trivial cha
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and therefore rk(V ) = (v(V ), (0,
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for some k ≫ 0. By the same argum
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and therefore Now consider the map
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In order to identify the kernel, no
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for all F , G stable sheave
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Chapter 6 Elliptic Calabi-Yau Three
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Definition 6.1.2. A projective morp
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Example 6.2.1. We’ve already seen
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contract 45 curves 2H+D-E contract
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Remark 6.3.2. This theorem basicall
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But this is impossible, because the
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We have H 0 (C, IQ) = 0 and χ(IQ)
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Having fixed a smooth point P ∈ C
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101 Proof. Since X
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103 Therefore, α can be represente
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Corollary 6.5.6. Under the hypothes
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107 Remark 6.6.4. We could also pro
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109 where the intersection form on
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111 where ci = ci(i∗U · ). Obvio
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113 One space of i
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Open Questions and Further Directio
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Bibliography [1] Aspinwall, P., Mor
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119 [28] D. Bayer and M. Stillman,
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derived categories of twisted sheaves on calabi-yau manifolds

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