derived categories of twisted sheaves on calabi-yau manifolds

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

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

Applying (ρ −1 i )∗ we get and thus (ρ −1 i )∗ O X k ij (ti − tj) = OXij (ρi(ti) − ρi(tj)) = OXij (si − τij(sj)) = OXij (si) −1 ⊗ OXij (τij(sj)) = OXij (si) ⊗ τ ∗ jiOXij (−sj) = OXij (si) ⊗ OXij (−sj) ⊗ L −1 ji = OXij (si) ⊗ OXij (−sj) ⊗ OXij ((k − 1)(si − sj)) = OXij (k(si − sj)), ϕ ∗ (µ k i ) ∗ O X k ij (ti − tj) = µ ∗ i (ρ −1 i )∗ O X k ij (ti − tj) = µ ∗ i OXij (k(si − sj)), which shows that if X is represented by α ∈ XS(JX), then X k is represented by (ϕ −1 ) ∗ α k in XS(J X k). The result about the existence <strong>of</strong> a universal sheaf is known (see, for example, [6]). But here is a quick pro<strong>of</strong> <strong>of</strong> the existence <strong>of</strong> a universal sheaf when (k, n) = 1: we only need to show that we can twist the universal <strong>sheaves</strong> described in the beginning <strong>of</strong> the pro<strong>of</strong> in a way that would make them glue. A computation similar to the one done in Section 4.3 shows that Mij = Uj|Pij ⊗ U −1 i |Pij is equal to the pull-back <strong>of</strong> a line bundle Fij from X k ij, and Fij can be taken to be <strong>of</strong> the form ρ ∗ i L −1 ij = ρ∗ i OXij ((k − 1)(sj − si)). Now let S be a line bundle on X, <strong>of</strong> fiber degree n (such a line bundle exists globally on X, by the comments at the end <strong>of</strong> the previous section), and consider ). We have the line bundles S ′ i = ρ ∗ i Si (where Si = S |Xi (ρ −1 j )∗ (S ′ j ⊗ (S ′ i ) −1 ) = Sj ⊗ τ ∗ ijS −1 i = Sj ⊗ S −1 i ⊗−n ⊗ L = OXij (−n(k − 1)(si − sj)). Therefore, considering the collections <strong>of</strong> line bundles {O X k i (ti)} and {S ′ i }, we see that their coboundaries are {O(k(si −sj))} and {O(−n(k −1)(si −sj))} (where we have omitted the subscript Xij and the pull-backs by isomorphisms to X k because these do not matter, since we are dealing with line bundles <strong>of</strong> degree 0). Since k and −n(k − 1) are coprime, we can obtain any multiple <strong>of</strong> {O(si − sj)} out <strong>of</strong> them, in particular we can construct {O((k − 1)(sj − si))}, which we have seen is the obstruction to gluing the Ui’s together. Therefore this obstruction is trivial, and hence the Ui’s can be glued to form a global universal sheaf, thus finishing the pro<strong>of</strong>. ij 60
Remark 4.5.3. Note that we have fixed a special isomorphism <strong>of</strong> JX and J X k in order to obtain this result. This is what allows us to distinguish between X and X −1 , which are isomorphic, even as fibrations over S! Indeed, one can take I∆ as a universal sheaf on X ×S X, where ∆ is the diagonal in X ×S X. But in this case, the isomorphism between JX and J X −1 is doing a negation along the fibers, which also acts by −1 on the Brauer group. Note that this answers the question posed at the end <strong>of</strong> [13, Section 1]. Another consequence <strong>of</strong> this remark is that if the j invariant is not constant (so that the only automorphisms <strong>of</strong> J/S are given by translations and negation), and k is not equal to 1 or −1 in Z/nZ, then X k is not isomorphic to X, even at the generic point <strong>of</strong> S. Thus if we can prove that there is only one elliptic fibration structure whose Jacobian is the same as that <strong>of</strong> X among all birational models <strong>of</strong> X, then X is not birational to X k . (Because, if they were birational to each other, they would have to be isomorphic to one another because they have the same Jacobian, and this is impossible.) See also Section 6.7. 61
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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 ∈
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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 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: It is easy to see that ϕi(ti) = [O
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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