derived categories of twisted sheaves on calabi-yau manifolds

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

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

Proposition 4.2.3. Assume that pX has a section s : S → X. Then J and X are isomorphic as schemes or analytic spaces over S; an isomorphism ϕ : X → J can be chosen in such a way that the natural section s0 <strong>of</strong> pJ (that corresponds to OX on X) is mapped to s. The moduli problem is fine, under these circumstances, and a universal sheaf can be taken to be OX×SJ(Γ) ⊗ π ∗ XOX(−s), where Γ is the (scheme theoretic) graph <strong>of</strong> ϕ inside X ×S J, πX : X ×S J → X is the projection, and OX(−s) is the line bundle on X defined by the divisor −s, the image <strong>of</strong> the section s : S → X. Pro<strong>of</strong>. This is nothing more than a relative version <strong>of</strong> the corresponding, well known statement for elliptic curves. This definition only works here for smooth elliptic fibrations (because Γ is a Cartier divisor). In the general case we’ll have to replace OX×SJ(Γ) with I ∨ Γ , the dual <strong>of</strong> the ideal sheaf <strong>of</strong> Γ (which makes sense always). This identification allows us to give an explicit construction <strong>of</strong> the Jacobian <strong>of</strong> X → S. For simplicity we’ll work in the analytic category. Cover S with open sets Ui such that if we set Xi = X ×S Ui, the projection Xi → Ui admits a section. For each i, fix once and for all a section si : Ui → Xi. We identify si with its image in Xi, whenever there is no danger <strong>of</strong> confusion. It is a divisor in Xi, whose restriction to each fiber has degree 1. We digress to talk about translations on elliptic curves. Let C be a smooth curve <strong>of</strong> genus 1 over an algebraically closed field (i.e., an elliptic curve, but without the choice <strong>of</strong> an origin). On C there exists a natural identification between line bundles <strong>of</strong> degree 0 and translations <strong>of</strong> C (under the group law on C that is obtained by fixing some origin), which is independent <strong>of</strong> the choice <strong>of</strong> an origin. This identification can be described as follows: let L ∈ Pic ◦ (C); fixing an origin s0 ∈ C, L can be written in a unique way as L ∼ = OC(s0 − p) for some p ∈ C. The translation associated to L , τL , is x ↦→ x + p for x ∈ C, where the operation is the one given in the group law <strong>of</strong> C, with origin fixed at s0. It is a trivial check to see that τL is in fact independent <strong>of</strong> the choice <strong>of</strong> s0. Conversely, given a translation τ, we can associate to it an element Lτ <strong>of</strong> Pic ◦ (C) by the formula Lτ = τ ∗ (F ) ⊗ F −1 , where F is some line bundle <strong>of</strong> degree 1 on C. This is independent <strong>of</strong> the choice <strong>of</strong> F , and the operations L ↦→ τL , τ ↦→ Lτ are inverse to one another. The important property <strong>of</strong> Lτ is the following: if F is a line bundle on C (not necessarily <strong>of</strong> degree 0), then we have τ ∗ F ∼ = F ⊗ L deg F τ A consequence <strong>of</strong> this formula is that if F ∈ Pic ◦ (C), then τ ∗ F ∼ = F , so if C ′ is isomorphic to C via a translation, there is a well defined notion <strong>of</strong> pull-back <strong>of</strong> F . 52
to C ′ , independent <strong>of</strong> the choice <strong>of</strong> translation: all pull-backs will give the same line bundle (up to isomorphism). We now return to the initial situation. Define Uij = Ui ∩ Uj, Xij = Xi ∩ Xj, and similarly Uijk, Xijk, etc. On each Xij there are two sections, si and sj, which define the line bundle Lij = OXij (sj − si), whose degree along the fibers <strong>of</strong> Xij → Uij is zero. As such it defines a translation τij : Xij → Xij over Uij. It is useful to note that τji = τ −1 ij , and τij ◦ τjk ◦ τki = id on Xijk. We think <strong>of</strong> these translations as gluing functions for a new space J, obtained from the “slices” Xi, glued along τij : Xj|Xi∩Xj → Xi|Xi∩Xj . Since the compatibility relations (checked above) hold, we can actually glue together this space, and it obviously comes with a natural map pJ : J → S which makes it into a smooth elliptic fibration. This construction naturally provides us with isomorphisms ρi : Ji → Xi over Ui, that satisfy ρi ◦ ρ −1 j = τij over Xij. The images <strong>of</strong> the sections si ⊆ Xi and sj ⊆ Xj, under ρ −1 i coincide in Jij: indeed, this is because we have ρi ◦ ρ −1 j (sj) = τij(sj) = sj − (sj − si) = si. 53 and ρ −1 j , (We have written here −(sj − si) for τij = τO(sj−si), to make things more evident.) Therefore the images <strong>of</strong> all the local sections under the local isomorphisms glue together in J to give a global section t0 <strong>of</strong> J. Now we can see how a point p ∈ J that lies over a point s ∈ S corresponds to a line bundle on Xs: let Ui be an open set such that s ∈ Ui, and associate to p the line bundle ρ ∗ i OJs(p − t0(s)). Since this line bundle has degree 0, the resulting line bundle on Xs does not change if we make another choice for the open set in which s lies, say Uj. In that case, we would obtain ρ ∗ jOJs(p − t0(s)), and the two line bundles differ by the pull-back by a translation <strong>of</strong> Xs. Since we are working with line bundles <strong>of</strong> degree 0, the isomorphism type <strong>of</strong> the result is not affected by this operation. 4.3 The Twisted Poincaré Bundle We consider the question <strong>of</strong> the existence <strong>of</strong> a universal sheaf on the product X ×S J, since in this setting we can explicitly construct (as a gerbe) the twisting α <strong>of</strong> Definition 3.3.3.
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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
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 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
Page 61: X which would intersect a general f
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 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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