derived categories of twisted sheaves on calabi-yau manifolds

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

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

Pro<strong>of</strong>. For the first statement, see [32, Remark A.7] and the description <strong>of</strong> <strong>twisted</strong> <strong>sheaves</strong> in the first chapter <strong>of</strong> this work. The other statements are [32, 6.4]. Note that n is defined in a slightly different fashion than in [32, 6.4], but this is the correct version <strong>of</strong> it: otherwise one could have, for example, a Mukai vector v = (r, l, s) for which gcd(r, s) = 1, but gcd(l.d) > 1 when d runs over all d ∈ NS(X). In this case, the pro<strong>of</strong> <strong>of</strong> [loc.cit.] fails unless one takes the correct value <strong>of</strong> n, which is 1. Theorem 5.1.14 (Orlov). Two K3 surfaces, X and M, have equivalent <strong>derived</strong> <strong>categories</strong>, D b coh (X) ∼ = D b coh (M), if and only if the lattices TX and TM are Hodge isometric. Pro<strong>of</strong>. See [36]. Here is a sketch <strong>of</strong> the pro<strong>of</strong>: when Db coh (X) ∼ = Db coh (M), the equivalence must be given by a Fourier-Mukai transform by Theorem 3.1.16 (this is the core <strong>of</strong> Orlov’s paper). Then, the results in Section 3.1 give Hodge isometries between TX and TM. In the other direction, one proves that if TX and TM are Hodge isometric, one can extend this isometry to an isometry i : ˜ H(X, Z) ∼ = ˜ H(M, Z), and this can be done in such a way that if v = i−1 (0, 0, ωM), one has on X a polarization such that M = M(v). Since n = 1 in this case (by Theorem 5.1.12), there exists a universal sheaf on X × M, which induces a Fourier-Mukai transform Db coh (X) ∼ = Db coh (M) (by Theorem 5.5.1). 5.2 Deformations <strong>of</strong> Twisted Sheaves In this section we consider what happens when one tries to deform a rank n vector bundle E0 on the central fiber X0 <strong>of</strong> a deformation, under the hypothesis that c1(E0) does not extend to an algebraic class on neighboring fibers. This analysis will be used for identifying the obstruction α from Definition 5.1.8. We are working over C. Let’s first set up the context. We start with f : X → S, a proper, smooth morphism <strong>of</strong> schemes or analytic spaces, and with 0 a closed point <strong>of</strong> S. The Brauer group we consider is Br ′ an(X)tors, which is the natural generalization to the analytic setting <strong>of</strong> the étale Brauer group used in the algebraic case. Throughout this section we’ll be loose in our notation and refer to Br ′ an(X)tors as the Brauer group <strong>of</strong> X, or Br ′ (X). Let X0 be the fiber <strong>of</strong> f over 0. We consider an element α ∈ Br(X), such that α|X0 is trivial as an element <strong>of</strong> Br(X0), and E a locally free α-<strong>twisted</strong> sheaf on X. Since α|X0 = 0, we can find a modification <strong>of</strong> the transition functions <strong>of</strong> E |X0 by a coboundary in H2 (X0, O∗ ) such that the transition functions glue, to get an X0 un<strong>twisted</strong> locally free sheaf E0 on X0. We want to understand what happens to c1(E0) in the neighboring fibers. Note that since the morphism f is smooth, by possibly restricting first the base S to a smaller open set (analytic or étale), we can identify the cohomology 68
groups <strong>of</strong> different fibers. Let’s explain this better (in the analytic cohomology, where things are simpler): since the morphism is smooth, the topology <strong>of</strong> X is necessarily that <strong>of</strong> X0 × S, after restricting the base enough to make it simply connected. Then the maps i ∗ s : H ∗ (X, Z) → H ∗ (Xs, Z) are isomorphisms, where is : Xs → X is the natural embedding <strong>of</strong> the fiber Xs <strong>of</strong> f over a point s ∈ S. The composition <strong>of</strong> one <strong>of</strong> these morphisms with the inverse <strong>of</strong> another gives a canonical way to identify the integral cohomology groups <strong>of</strong> the fibers <strong>of</strong> X. It is worthwhile observing that for any space X we have H 2 (X, Q/Z) = H 2 (X, Z) ⊗ Q/Z ⊕ H 3 (X, Z)tors, from the universal coefficient theorem. We saw (Theorem 1.1.3) that Br ′ (X) is the quotient <strong>of</strong> H 2 (X, Q/Z) by the image <strong>of</strong> Pic(X)⊗Q/Z. But classes in H 3 (X, Z)tors cannot become zero in this quotient, so (in view <strong>of</strong> the fact that cohomology groups <strong>of</strong> the fibers are locally constant over the base) the only way an element <strong>of</strong> Br ′ (X) can become zero in a central fiber X0 without being zero in the neighboring fibers is if it belongs to H 2 (X, Z) ⊗ Q/Z, and in the central fiber it is also in the image <strong>of</strong> Pic(X0) ⊗ Q/Z. This situation (which is typical for, say, K3 surfaces, for which H 3 (X, Z)tors = 0) should be contrasted with the situation for Calabi-Yau threefolds. The two cases are quite opposite, when it comes to deformations: in the case <strong>of</strong> K3 surfaces, we can have a class in H 2 (X, Q/Z) which becomes zero in the Brauer group <strong>of</strong> a central fiber, without being zero in the neighboring fibers, by “suddenly becoming algebraic”. In the case <strong>of</strong> Calabi-Yau threefolds, this cannot happen, since we are only dealing with classes in H 3 (X, Z)tors. However, it can happen that all neighboring fibers Xs, s = 0, have trivial Brauer group, while the central fiber X0 has a non-trivial Brauer group, by having a non-torsion class in H 3 (Xs, Z) become torsion in H 3 (X0, Z). Obviously, in this case the morphism X → S can not be smooth. The reader should consult Chapter 6 for more details, and for an example when this phenomenon occurs. We now return to the situation we started studying in the beginning <strong>of</strong> this section, that <strong>of</strong> a smooth morphism f : X → S and an element α ∈ Br(X). Because <strong>of</strong> the previous considerations, assume that α belongs to H 2 (X, Z)⊗Q/Z, and therefore it can be written as 1 n c, for some class c ∈ H2 (X, Z). The fact that α|X0 = 0 means that c|X0 belongs to Pic(X0). We’d like to identify c|X0 (which is the same as finding c, because <strong>of</strong> the existence <strong>of</strong> the isomorphism H ∗ (X, Z) ∼ = H ∗ (X0, Z)). This is the content <strong>of</strong> the following theorem: Theorem 5.2.1. Let E be a locally free α-<strong>twisted</strong> sheaf on X, and let E0 = E |X0. Assume that S is small enough (say, contractible), so that we have an identification H 2 (X, Z) ∼ = H 2 (X0, Z). Since α|X0 = 0, we can view E0 as an usual sheaf on X0, by Lemma 1.2.8. Then α = − 1 rk(E0) c1(E0), 69
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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.
Page 11 and 12:
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
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 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: Theorem 5.1.10. Under the hypothese
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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