derived categories of twisted sheaves on calabi-yau manifolds

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

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

5.5 Relationship to Derived Categories . . . . . . . . . . . . . . . . . . 82 6 Elliptic Calabi-Yau Threefolds 85 6.1 Generic Elliptic Calabi-Yau Threefolds . . . . . . . . . . . . . . . . 86 6.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.3 Jacobians <strong>of</strong> Curves <strong>of</strong> Genus 1 . . . . . . . . . . . . . . . . . . . . 92 6.4 The Relative Jacobian . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.5 The Twisted Pseudo-Universal Sheaf and Derived Equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.6 Equivalent Twistings . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.7 Applications to the Torelli Problem . . . . . . . . . . . . . . . . . . 108 6.8 Relationship to Work <strong>of</strong> Aspinwall-Morrison and Vafa-Witten . . . 112 III Open Questions and Further Directions 115 Bibliography 117 vii
List <strong>of</strong> Figures 6.1 The Kähler cones in Example 6.2.2 . . . . . . . . . . . . . . . . . . 90 6.2 The Kähler cones in Example 6.2.3 . . . . . . . . . . . . . . . . . . 91 viii
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: Table of Contents
Page 11 and 12: Introduction and Overview Twisted <
Page 13 and 14: twisted by α ∈
Page 15 and 16: fibers introduces interesting featu
Page 17 and 18: Chapter 1 Twisted Sheaves In this c
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
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X which would intersect a general f
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to C ′ , independent of</
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Jij → S. This means that if Fij w
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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
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allow us to get X; however, finding
Page 75 and 76:
endowed with the product ((r, l, s)
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Theorem 5.1.10. Under the hypothese
Page 79 and 80:
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
Page 99 and 100:
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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