derived categories of twisted sheaves on calabi-yau manifolds

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

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

The next step is to understand this picture further, and using the Leray spectral sequence one shows that if the fibrations in question have no singular fibers there is an exact sequence 0 → Br(S) −→ Br(J) π −→ XS(J) → 0 which splits. (The first map is just the pull-back map on the Brauer groups.) The splitting is provided by, say, the pull-back from J to S via a section <strong>of</strong> J → S. Let’s see how this relates to the situation described in the previous section: the construction <strong>of</strong> the gerbe α provides an element <strong>of</strong> Br(J), and projecting we obtain an element π(α) <strong>of</strong> XS(J). In fact, the line bundles ρ ∗ i OXij (si − sj) can be written in a unique way as OJij (tij − t0) for some tij ∈ J # (Uij), and it is clear from the construction that in order to reconstruct X → S one needs to reglue the Ji’s via the translations τ −1 ij = τOJ ij (tij−t0) which are the inverses <strong>of</strong> the translations τij used before to construct J from X. But the translations {τ −1 ij } give the element αX <strong>of</strong> XS(J) that represents X (see the construction described above), and tracing through all the identifications it is easy (!) to see that αX is precisely the projection π(α) <strong>of</strong> α. Therefore we obtain: Theorem 4.4.1. Let X → S be a smooth elliptic fibration, and let J → S be its relative Jacobian. Let αX ∈ XS(J) be the element <strong>of</strong> the Tate-Shafarevich group that represents X → S, and let α ∈ Br(J) be the obstruction to the existence <strong>of</strong> a universal sheaf on X ×S J (for the moduli problem on X that gives J). Then αX = π(α), where π is the projection 0 → Br(S) −→ Br(J) π −→ XS(J) → 0. Theorem 4.4.2. Let n denote the smallest positive fiber degree <strong>of</strong> a divisor on X. (The fiber degree <strong>of</strong> a divisor D is defined as the degree <strong>of</strong> the restriction D|F <strong>of</strong> D to a fiber <strong>of</strong> X → S.) Then the projection αX = π(α) is n-torsion in XS(J). Pro<strong>of</strong>. Since we work in the Tate-Shafarevich group we can ignore contributions from the Brauer group <strong>of</strong> the base, and we only need to prove that there exist line bundles Li on Ji such that F ⊗n ij ∼ = Li ⊗ L −1 j on Jij. (This proves that, up to contributions from Br(S), the gerbe {Fij} defined in the previous section is trivial.) Let D be a divisor on X <strong>of</strong> fiber degree n, and let L = OX(D). Define 56
Li = ρ∗ i L −1 |Xi . Then we have Li ⊗ L −1 j = ρ∗ i L −1 ⊗ ρ ∗ jL = ρ ∗ i (L −1 ⊗ (ρ −1 i )∗ ρ ∗ jL ) = ρ ∗ i (L −1 ⊗ τ ∗ jiL ) = ρ ∗ i (L −1 ⊗ L ⊗ L ⊗n τji ) = ρ ∗ i (L −1 ⊗ L ⊗ L ⊗n ji ) = ρ ∗ i OX(si − sj) ⊗n = F ⊗n ij , where we have used the comments in Section 4.2 regarding translations <strong>of</strong> line bundles <strong>of</strong> various degrees. (Note that we have used the fact that deg L = n.) Remark 4.4.3. In fact, using some slightly more detailed information that one can get from Ogg-Shafarevich theory it can be proven that the order <strong>of</strong> αX in XS(J) is precisely n. 4.5 Other Fibrations The Jacobian fibration that we considered in the previous sections is not the only one we can naturally construct from the original fibration X → S. In a certain sense the Jacobian is the common denominator <strong>of</strong> the other ones, but these present some interest as well, and we study them briefly here. We keep the notation from the previous sections. Definition 4.5.1. For any integer k, let Y → S be the relative moduli space <strong>of</strong> semistable <strong>sheaves</strong> <strong>of</strong> rank 1, degree k on the fibers <strong>of</strong> X → S. We call Y → S the k-th <strong>twisted</strong> power <strong>of</strong> X → S, and denote it by X k → S. Theorem 4.5.2. For any integer k, X k → S is a smooth elliptic fibration whose relative Jacobian is isomorphic in a natural way to the relative Jacobian J → S <strong>of</strong> the initial fibration X → S. If α ∈ XS(J) is the element <strong>of</strong> the Tate-Shafarevich group representing X → S, then X k → S is represented by α k . Consequently the fibration (X k ) k′ → S is isomorphic to X kk′ XS(J). A universal sheaf exists on X ×S X k if and only if (k, n) = 1, 57 → S. Let n denote the order <strong>of</strong> α in and in this case X can be viewed as a moduli space on X k (we have X ∼ = (X k ) k′ as fibrations over S, where k ′ is any integer such that kk ′ = 1 mod n).
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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 and 62: X which would intersect a general f
Page 63 and 64: to C ′ , independent of</
Page 65: Jij → S. This means that if Fij w
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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