derived categories of twisted sheaves on calabi-yau manifolds

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

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

We continue with the notations and assumptions from the previous section. Let Pi = Xi ×S Ji, and similarly Pij, etc. Note that over each Pi we have a universal sheaf given by Ui = OX×SJ(Γi) ⊗ π ∗ XOXi (−si), as in Proposition 4.2.3. (We abuse the notation and keep using πX for the projection Xi → Ui.) The natural question to ask at this point is why don’t these local universal <strong>sheaves</strong> glue to a global one? We need to look over an intersection Pij, and see what is the failure <strong>of</strong> Ui|Pij and Uj|Pij to be isomorphic. Let Mij = Uj|Pij ⊗ U −1 i |Pij , an operation which we can do in this case since we are dealing with line bundles; we have Mij = π ∗ XOXi (si − sj) ⊗ O(Γj) ⊗ O(Γi) −1 . Let p be any point in Jij, let s be its image in S, and consider the restriction <strong>of</strong> Mij to Xs. It is isomorphic to OXs(si − sj + ρj(p) − ρi(p)), which is trivial along Xs (we could have seen this also from the fact that Ui and Uj are local universal <strong>sheaves</strong>, so over p ∈ Jij they must give the same line bundle on Xs, the one represented by p as a point <strong>of</strong> the moduli space J). We conclude that Mij is the pull-back <strong>of</strong> a line bundle Fij from Jij. The collection {Fij}, viewed as a gerbe α ∈ H 2 (J, O ∗ J 54 ) will represent the twisting that is the obstruction to the existence <strong>of</strong> a universal sheaf on X ×S J. It is worthwhile to note that the twisting π ∗ J α is in fact trivial on X ×S J: indeed, the collection {Uij} is a π ∗ J α-<strong>twisted</strong> line bundle on X ×S J, so α must have order divisible by 1 (the rank <strong>of</strong> this vector bundle). In other words α must be trivial. The point here is that although we could modify this line bundle to make it glue well on X ×S J, this operation would not preserve the fibers <strong>of</strong> this bundle on fibers <strong>of</strong> πJ, so the result would no longer be a universal sheaf. It is not hard to see that the isomorphism type <strong>of</strong> Fij (at least along the fibers <strong>of</strong> Jij → S) can be described as follows: if on Xij we consider the sheaf L −1 ij = OXij (si − sj), then Fij is isomorphic to the sheaf ρ ∗ i L −1 ij . The remarks in the previous section regarding the invariance <strong>of</strong> line bundles <strong>of</strong> degree zero under translations show that this gives a well-defined isomorphism type for Fij. An easy verification shows that } satisfies the conditions to form a gerbe. the collection {ρ∗ i L −1 ij However, this construction is not entirely natural: we have made the choice <strong>of</strong> using ρi as an isomorphism between Jij and Xij. In fact, the verification that we can do to ensure that {Fij} is indeed a gerbe only tests this along the fibers <strong>of</strong>
Jij → S. This means that if Fij were <strong>twisted</strong> by line bundles from Uij, we would not be able to detect this. To make this construction independent <strong>of</strong> choices, one could consider the <strong>sheaves</strong> Fij = πJ,∗Mij. These only differ from the Fij’s constructed before by the global line bundle πJ,∗OX×SJ, and therefore the gerbe determined by {Fij} is the one we want. Under this definition <strong>of</strong> Fij it is obvious that the compatibility relations are satisfied. This situation should be contrasted to the one in the next section, where Ogg- Shafarevich theory produces a natural element <strong>of</strong> Br(J)/ Br(S). This should be thought <strong>of</strong> as meaning that by just looking at the picture along the fibers (which is what we did in this section) we can only recover α up to possible twists from Br(S). 4.4 Ogg-Shafarevich Theory As alluded to at the end <strong>of</strong> the previous section, this picture has a strong relationship to the classification theory for elliptic fibrations, as developed by Ogg and Shafarevich. For a reference to the technical aspects <strong>of</strong> this theory, see [13]. The setup is as before: X → S is a smooth elliptic fibration, and J → S is its relative Jacobian fibration. Since X → S is obtained from J → S by the “cut and reglue” procedure described before, one can try to understand this process from a cohomological point <strong>of</strong> view. One approach is to note that the regluing data consists <strong>of</strong> translations along the fibers <strong>of</strong> each Ji; the group <strong>of</strong> these translations can be identified with the group J # (Ui) <strong>of</strong> sections <strong>of</strong> Ji → Ui (which can be naturally given the structure <strong>of</strong> a group because we have a fixed, natural section t0 <strong>of</strong> J → S). Therefore the data needed to reconstruct the fibration X → S out <strong>of</strong> J → S is an element αX <strong>of</strong> the Tate-Shafarevich group <strong>of</strong> J, XS(J), defined as XS(J) = H 1 (S, J # ), where J # is the sheaf <strong>of</strong> abelian groups on S defined by J # (U) = the group <strong>of</strong> sections <strong>of</strong> JU → U. This makes sense in any topology, and if one considers everything in the analytic or étale topology, one obtains a 1-1 correspondence between elliptic fibrations X → S whose Jacobian is J → S, and elements <strong>of</strong> XS(J). (It is important to note here that some fibrations that correspond to elements <strong>of</strong> H 1 an(S, J # ) can only be constructed in the analytic category and have no algebraic counterpart; to remain inside the algebraic category one has to work with in the étale setup, and thus obtain fewer elliptic fibrations.) 55
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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: to C ′ , independent of</
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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