derived categories of twisted sheaves on calabi-yau manifolds

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

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

nomenon observed in Section 5.5: on ¯ J, we get an equivalence <strong>of</strong> <strong>derived</strong> <strong>categories</strong> D b coh( ¯ J, α) ∼ = D b coh( ¯ J, α k ), where α ∈ Br( ¯ J) is the twisting associated to X via Ogg-Shafarevich theory, and gcd(k, ord(α)) = 1. In the next section we discuss the relationship <strong>of</strong> <strong>derived</strong> <strong>categories</strong> and the Torelli problem for Calabi-Yau threefolds. We show that Calabi-Yau threefolds with equivalent <strong>derived</strong> <strong>categories</strong> have the same Hodge data, and hence they provide a counterexample to the generalization <strong>of</strong> Torelli from K3 surfaces. In particular, we provide such an example based on the elliptic Calabi-Yau’s studied in Section 6.2. We finish this chapter with a section on the relationship <strong>of</strong> these examples with the example <strong>of</strong> Vafa-Witten ([43]) and Aspinwall-Morrison-Gross ([1]). We give a quick overview <strong>of</strong> their example, and suggest a possible explanation for the occurrence <strong>of</strong> stable singularities in string theory. 6.1 Generic Elliptic Calabi-Yau Threefolds In this section we define the objects <strong>of</strong> study in this chapter: a certain class <strong>of</strong> elliptic Calabi-Yau three-folds, in which we have very good control over the degenerations <strong>of</strong> the elliptic fibers. The results in this section could probably be easily generalized to arbitrary elliptic fibrations without multiple fibers, but since the results here are intended only to give life to some large class <strong>of</strong> examples, we’ll limit ourselves to this sufficiently general situation. Recall the definition <strong>of</strong> a Calabi-Yau threefold: Definition 6.1.1. A smooth, projective algebraic variety or analytic space X is called a Calabi-Yau threefold if it has (complex) dimension 3, and if it satisfies KX = 0 and H 1 (X, OX) = 0. (Here, and in the future, KX denotes the canonical divisor <strong>of</strong> X.) We will be focused on studying the case <strong>of</strong> complex threefolds, in which case the Hodge structure <strong>of</strong> X will come into play. It is described by the Hodge diamond 1 0 0 0 h 2,2 1 h 2,1 h 1,2 0 h 1,1 0 0 1. Of course, h 1,1 = h 2,2 , and h 2,1 = h 1,2 . The following definitions are from [13]: 0 0 1 86
Definition 6.1.2. A projective morphism f : X → S <strong>of</strong> algebraic varieties or analytic spaces is called an elliptic fibration if its generic fiber E is a regular curve <strong>of</strong> genus one and all fibers are geometrically connected. If X is a Calabi-Yau threefold, then f is called an elliptic Calabi-Yau threefold. Remark 6.1.3. Most <strong>of</strong> the time we’ll abuse the notation, by saying “let X be an elliptic Calabi-Yau threefold.” By this we mean “let X be a Calabi-Yau threefold, and let f : X → S be an elliptic fibration on X.” Definition 6.1.4. Let f : X → S be an elliptic fibration. The locus ∆ ⊆ S <strong>of</strong> points s ∈ S such that f is not smooth at some point x ∈ X with f(x) = s, is called the discriminant locus <strong>of</strong> f. It is a Zariski closed subset <strong>of</strong> S. The closed subset <strong>of</strong> ∆ ∆ m = {s ∈ S | f is not smooth at any x ∈ f −1 (s)} is called the multiple locus <strong>of</strong> f. A fiber over a point s ∈ ∆ m is called a multiple fiber. We will only be interested in studying elliptic fibrations without multiple fibers, so from now on we make the assumption that ∆ m = ∅. Definition 6.1.5. A section (resp. a rational section) <strong>of</strong> an elliptic fibration f : X → S is a closed subscheme Y <strong>of</strong> X for which the restriction <strong>of</strong> f to Y is an isomorphism (resp. a birational morphism). A degree n multisection (or an nsection) is a closed subscheme Y <strong>of</strong> X such that the restriction <strong>of</strong> f to Y is a finite morphism <strong>of</strong> degree n. A rational degree n multisection is a closed subscheme Y <strong>of</strong> X such that the restriction <strong>of</strong> f to Y is generically finite <strong>of</strong> degree n. When working in the algebraic category, it is well-known that any elliptic fibration has a rational multisection. In the analytic category this does not necessarily hold true any more, but we will always assume not only the existence <strong>of</strong> a rational multisection, but even more: that the elliptic fibrations under consideration have multisections (not only rational). Definition 6.1.6. For the rest <strong>of</strong> this chapter, define a generic elliptic Calabi-Yau threefold to be an elliptic threefold f : X → S, with X and S smooth algebraic varieties over C or complex manifolds, X Calabi-Yau, and satisfying the following extra properties: 1. f is flat (i.e. all fibers are 1-dimensional); 2. f does not have any multiple fibers; 3. f admits a multisection; 4. the discriminant locus ∆ is a reduced, irreducible curve in S, having only nodes and cusps as singularities; 87
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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
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 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: Chapter 6 Elliptic Calabi-Yau Three
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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