derived categories of twisted sheaves on calabi-yau manifolds

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

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

114 into account also <strong>derived</strong> <strong>categories</strong> <strong>of</strong> <strong>twisted</strong> <strong>sheaves</strong>. This is rather vague, but it seems to coincide with physicists’ belief that when the Brauer group <strong>of</strong> a space is non-trivial, one can construct more than one physical theory on that space, by introducing into the game the elements <strong>of</strong> the Brauer group. One can even attempt to pursue this one step further: by work <strong>of</strong> Bridgeland- King-Reid ([8]), it is known that one has D G coh(Z) ∼ = D b coh(Y ), where Y is the crepant resolution we constructed before. Here, DG coh (Z) denotes the bounded <strong>derived</strong> category <strong>of</strong> coherent, G-equivariant <strong>sheaves</strong> on Z. The group H2 (G, C∗ ) classifies central extensions <strong>of</strong> the form 0 → C ∗ → K → G → 0. If one considers the trivial extension, K = G ⊕ C∗ , one obtains that the Kequivariant <strong>sheaves</strong> on Z are just the G-equivariant <strong>sheaves</strong> on Z. However, when one considers the non-trivial extension K (which corresponds to the non-trivial element <strong>of</strong> H2 (G, C∗ )), one obtains a new <strong>derived</strong> category, DK coh (Z). This seems to suggest that one should expect an equivalence <strong>of</strong> the sort D K coh(Z) ∼ = D b coh(X, α), where X is any deformation <strong>of</strong> Z having ODP’s at the 64 prescribed points, and α is the unique non-trivial element <strong>of</strong> Br(X). Of course, this cannot be expected to hold as such. Probably, we would have to take a small, analytic resolution <strong>of</strong> the singularities <strong>of</strong> X, as we did in the case <strong>of</strong> the relative Jacobian. Also, another problem with this equivalence is that it seems to be independent <strong>of</strong> the particular deformation taken. The fact that X and Y are mirrors seems to suggest that D b coh (X, α) should in fact be equivalent to some Fukaya category on Z. The relationship <strong>of</strong> this situation to the one we studied before is the following: if X is the elliptic Calabi-Yau X that we considered in the previous sections, let J be its relative Jacobian. On ¯ J, a small resolution <strong>of</strong> J, we have two interesting, and entirely different, <strong>derived</strong> <strong>categories</strong>: one, Db coh ( ¯ J, 0) is the usual <strong>derived</strong> category <strong>of</strong> ¯ J. The other one, Db coh ( ¯ J, α), is the same as Db coh (X). If <strong>derived</strong> <strong>categories</strong> are thought <strong>of</strong> as physical theories, this means that on ¯ J we can construct two different (X) and physical theories, both <strong>of</strong> them “smooth” (since they correspond to Db coh Db coh ( ¯ J), and both spaces are smooth). However, if one attempts to deform J so as to remove its singularities, one observes that its Brauer group becomes zero. Thus, from a deformed J, one cannot hope to recover more than one physical theory (Db coh (J)), a situation very similar to the one studied by Vafa and Witten.
Open Questions and Further Directions In this chapter we collect everything that was left loose-ended in the other chapters. Its motto should be “a conclusion is the place where you stopped because you were tired <strong>of</strong> thinking.” First, there are technical issues whose solution is quite unsatisfactory. In this category I would list the following: 1. One would like to be able to deal confidently with <strong>derived</strong> <strong>categories</strong> <strong>of</strong> <strong>twisted</strong> <strong>sheaves</strong> even when the twisting is not in the Brauer group. In particular one would like to understand the boundedness <strong>of</strong> <strong>derived</strong> functors in this context. 2. There seems to be no particular reason why duality for proper morphisms should not hold in general for <strong>twisted</strong> <strong>sheaves</strong>. 3. Condition (TD) in Section 3.1 seems unnecessary. A correct definition <strong>of</strong> the Mukai vector and <strong>of</strong> the bilinear pairing should eliminate this restriction. 4. The requirement that there exist X and X ′ in Theorem 5.5.2 appears superfluous. Conjecture 5.5.3 should hold in general. 5. All the results in Chapter 6 should hold for arbitrary elliptic Calabi-Yau threefolds (no restrictions on the singularities <strong>of</strong> the fibers). 6. In particular, one should have D b coh (X, α) ∼ = D b coh (X, αk ) for any elliptic Calabi-Yau threefold X, and any α ∈ Br(X) with ord(α) and k coprime. In the second category I would list open questions whose solution I would have liked to be able to find, but I was not: 1. What is the relationship <strong>of</strong> the group Br(X)/ Aut(X) to the group <strong>of</strong> Morita equivalence classes <strong>of</strong> <strong>sheaves</strong> <strong>of</strong> Azumaya algebras on a scheme X? (Here we have considered the action <strong>of</strong> the automorphism group <strong>of</strong> X, Aut(X), on Br(X).) Are they the same? 115
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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
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So assume n > 0, and as before cons
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Proposition 2.3.4. Let f : X → Y
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2.4 Duality for Proper Smooth Morph
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complex manifolds, and let X × Y
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if v consists only of</stro
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The reason for the notation is that
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Proof. The first s
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which by definition associates to a
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Part II Applications 49
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X which would intersect a general f
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to C ′ , independent of</
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Jij → S. This means that if Fij w
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Li = ρ∗ i L −1 |Xi . Then we h
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It is easy to see that ϕi(ti) = [O
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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: 113 One space of i
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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