derived categories of twisted sheaves on calabi-yau manifolds

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

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

102 intersects the component <strong>of</strong> Xt that is not hit by si. (Each section intersects precisely one component <strong>of</strong> Xt, and we’re just switching components.) One can do this, by possibly restricting to a smaller open set, because Xt is reduced. Moving to a different component Xt corresponds to performing a flop to the map Xi → Ji, and therefore we get an isomorphism (over Ui) Xi ∼ = ¯ Ji. 6.5 The Twisted Pseudo-Universal Sheaf and Derived Equivalences Let f : X → S be a generic elliptic Calabi-Yau threefold, and let J → S be its relative Jacobian. Let ∆ be the discriminant locus <strong>of</strong> X → S, let U = S \ ∆, and consider the smooth elliptic fibration XU → U. Of course, JU → U is the relative Jacobian <strong>of</strong> XU → U, and we obtain by the results in Chapter 4 a unique α ∈ Br(JU) that corresponds to XU → U (as an obstruction to the existence <strong>of</strong> a universal sheaf on XU ×U JU). Let p2 : X ×S J → J and ¯p2 : X ×S ¯ J → ¯ J be the projections. Theorem 6.5.1. There exists a unique α ′ ∈ H2 an(J, O∗ J ), whose restriction to JU equals α. For any analytic small resolution ρ : ¯ J → J <strong>of</strong> the singularities <strong>of</strong> J, let ¯α = ρ∗α. Then 1. ¯α ∈ Br( ¯ J); 2. there exists a ¯p ∗ 2 ¯α-sheaf U on X ×S ¯ J, flat over ¯ J, whose restriction to XU ×U ¯ JU = XU ×U JU is isomorphic to the p ∗ 2α-<strong>twisted</strong> universal sheaf <strong>of</strong> Proposition 3.3.2. Pro<strong>of</strong>. Throughout this pro<strong>of</strong>, let J s denote the stable part <strong>of</strong> J (which is just the smooth part <strong>of</strong> J), and ¯ J s = ρ −1 (J s ). Obviously, ¯ J s ∼ = J s . Sometimes we’ll identify ¯ J s with J s , when there is no danger <strong>of</strong> confusion. Using Theorem 6.4.6, find a covering {Ui} <strong>of</strong> S by analytic open sets, that satisfies the conditions in the pro<strong>of</strong> <strong>of</strong> that theorem, and isomorphisms ϕi : Xi → ¯ Ji <strong>of</strong> analytic spaces over Ui, where Xi = XUi , and similarly for Ji, ¯ Ji. Let Ui be the pull-back by id ×Ui ϕ−1 i to Xi ×S ¯ Ji <strong>of</strong> the family constructed in Proposition 6.4.2. Over ¯ J s , the <strong>sheaves</strong> Ui are local universal <strong>sheaves</strong>: indeed, this follows from the very way the maps ϕi is constructed, and the fact that ϕi is an isomorphism between Xi − Ci and Ji − xi when there is a curve Ci to be contracted to an ODP xi, and an isomorphism between Xi and Ji otherwise. Therefore, restricting to ¯ J s , the collection {Ui| J¯ s} forms an α-<strong>twisted</strong> universal sheaf.
103 Therefore, α can be represented on the cover {J s ∩ Ji}. Indeed, over each open set in this cover we have a universal sheaf (Ui|Js), and therefore we can construct α as in Proposition 3.3.2. Note that if i = j, Ji ∩ Jj ⊆ J s . Thus, if we set α ′ iii = 1, and α ′ ijk = αijk when not all three <strong>of</strong> i, j, k are equal, we get a well defined element α ′ <strong>of</strong> ˇ H 2 (J, O ∗ J ), whose restriction to J s is α. (The conditions for being a cocycle have to be verified only along fourfold intersections, and there they are already satisfied by α.) Note that this α ′ is obviously unique. The same situation holds for ¯ J: if i = j, ¯ Ji ∩ ¯ Jj ⊆ ¯ J s by the very construction <strong>of</strong> ¯ Ji and <strong>of</strong> the cover {Ui}. Since the only conditions for a collection <strong>of</strong> <strong>sheaves</strong> and isomorphisms to form a <strong>twisted</strong> sheaf happen on the intersections <strong>of</strong> the open sets in the cover, and in our case all these intersections lie inside ¯ J s , we conclude that {Ui} form a ¯p ∗ 2 ¯α-<strong>twisted</strong> sheaf on ¯ J, where ¯α = ρ ∗ α. (Here we could have used the standard purity theorem for cohomological Brauer groups, [18, III, 6.2].) The fact that ¯α is in fact in Br( ¯ J) follows immediately from Proposition 3.3.4. Remark 6.5.2. This result should be compared with the following analysis: by looking at the smooth part, the initial fibration corresponds to an element α ∈ XU(JU). Since there are no multiple fibers involved, α is actually in XS(J). One can blow up S to a new base S1, and pull-back J over S1 to get a resolution J1 <strong>of</strong> J. By [13, 2.18], we have an inclusion XS(J) ⊆ XS1(J1), and using [13, 1.17], we have XS1(J1) = Br ′ (J1)/ Br ′ (S1). Using an analytic form <strong>of</strong> [18, III, 7.3], we have Br ′ (J1) = Br ′ an( ¯ J)tors Br ′ (S1) = Br ′ an(S)tors. Since X is an elliptic Calabi-Yau threefold, S must be a rational or Enriques surface ([17, 2.3]), and thus we have Br ′ (S) = Br ′ an(S)tors = 0. We conclude that the element α ∈ XS(J) can be viewed as an element in XS1(J1) = Br ′ an( ¯ J)tors via Ogg-Shafarevich theory. This element coincides with α ′ by the analysis in Chapter 4, and the above theorem claims that it is actually an element <strong>of</strong> the Brauer group (i.e. it is represented by an Azumaya algebra). Theorem 6.5.3. Let U 0 be the extension by zero (push-forward by the inclusion map) <strong>of</strong> the ¯p ∗ 2 ¯α-sheaf U defined in the previous theorem, from X ×S ¯ J to X × ¯ J. (U 0 is naturally a (p ′ 2) ∗ ¯α-<strong>twisted</strong> sheaf on X × ¯ J, where p ′ 2 : X × ¯ J → ¯ J is the projection.) Then the Fourier-Mukai transform D(X) → D( ¯ J, ¯α) determined by U 0 is an equivalence <strong>of</strong> <strong>categories</strong>. Pro<strong>of</strong>. For P ∈ ¯ J, let F (P ) be the fiber <strong>of</strong> U over X ×S {P }, and let F 0 (P ) be the extension by zero <strong>of</strong> F (P ) to X (which is also the fiber <strong>of</strong> U 0 over X × {P }). Using the criterion <strong>of</strong> Theorem 3.2.1, and remarking that U 0 is flat over ¯ J, all we need to check is that
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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
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 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: 101 Proof. Since X
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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