derived categories of twisted sheaves on calabi-yau manifolds

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

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

Remark 5.5.4. There is one more thing to note here: having started with M and α, an easy way to find X would be to consider Db coh (M, α), and take X to be the “spectrum” <strong>of</strong> this <strong>derived</strong> category (see [39] for details on the spectrum <strong>of</strong> a triangulated category and related topics). This would work, <strong>of</strong> course, only if X existed, and by Theorem 5.4.3 we know that in this case TX = Ker(α). This would mean in particular that Ker(α) can be primitively embedded in LK3. This does not in general hold, but Theorem 5.4.3 gives us anyway a candidate for TX! The conclusion <strong>of</strong> this is that Db coh (M, α) should be viewed as a <strong>derived</strong> category <strong>of</strong> a “virtual” K3 X, where X itself might not necessarily exist, but it makes sense to speak <strong>of</strong> its transcendental lattice. Another interesting conjecture which relates to these topics is the following: Conjecture 5.5.5. If X and X ′ are K3 surfaces with D b coh (X) ∼ = D b coh (X′ ), then Theorem 5.4.3 gives us a natural isomorphism ϕ : Br(X) ∼ = Br(X ′ ). Is it then the case that D b coh (X, α) ∼ = D b coh (X′ , ϕ(α))? 84
Chapter 6 Elliptic Calabi-Yau Threefolds In this chapter we pursue further the results <strong>of</strong> Chapter 4, to the case <strong>of</strong> some elliptic fibrations with singular fibers. We are mainly interested in elliptic Calabi- Yau threefolds, and we only consider a number <strong>of</strong> generic examples here. By “generic” we mean that the singularities <strong>of</strong> the fibers are the simplest possible to still get a Calabi-Yau: the discriminant locus ∆ is a curve with only nodes and cusps as singularities, and the curve over the generic point <strong>of</strong> ∆ is a rational curve with one node (I1). (See Section 6.1 for details on this topic.) We provide three examples <strong>of</strong> such fibrations in Section 6.2. The first one is folklore, but the second and third ones are, to the best <strong>of</strong> my knowledge, new. Given a fibration with the above properties, our first concern is understanding the relative Jacobian, which is defined as a relative moduli space (Section 6.4). In order to do this, we need to understand moduli spaces <strong>of</strong> semistable <strong>sheaves</strong> on the fibers <strong>of</strong> X, and we do this in Section 6.3. Having done that, we go on to study the relative Jacobian in Section 6.4. We note here an interesting phenomenon: although the space X we start with is smooth, the relative Jacobian has singularities. These singularities can not be removed without losing the Calabi- Yau property (J has trivial canonical bundle, but a small – or crepant – resolution does not exist in general). The existence <strong>of</strong> these singularities would seem to prevent us from finding an equivalence <strong>of</strong> <strong>derived</strong> <strong>categories</strong> (since <strong>derived</strong> <strong>categories</strong> “detect smoothness”, as well as triviality <strong>of</strong> canonical bundle), but by working in the analytic category we are able to bypass this problem. Indeed, in this category we find a small resolution ¯J <strong>of</strong> the singularities <strong>of</strong> J, and we are able to show that there exists a <strong>twisted</strong> “pseudo-universal” sheaf on X ×S ¯ J (where S is the base <strong>of</strong> the fibrations). This <strong>twisted</strong> pseudo-universal sheaf is equal to the usual <strong>twisted</strong> universal sheaf over the stable part <strong>of</strong> ¯ J (which equals the stable part <strong>of</strong> J), but also parametrizes some <strong>of</strong> the semistable <strong>sheaves</strong> on X, over the properly semistable part <strong>of</strong> ¯ J (which is the resolution <strong>of</strong> the singular, semistable points <strong>of</strong> J). Using this pseudo-universal sheaf, we can define an integral transform, which turns out to be an equivalence. This is done in Section 6.5. We then note, in Section 6.6 another occurrence <strong>of</strong> the same surprising phe- 85
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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: for all F , G stable sheave
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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