derived categories of twisted sheaves on calabi-yau manifolds

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

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

QQ|X. After a fixed twist <strong>of</strong> one <strong>of</strong> them by a line bundle, it can be arranged that they have the same Chern classes, call them c0, c1, c2. Mukai proves that these vector bundles are stable, and that they are all different for different quadrics (and among themselves). However, when the quadric becomes singular, the two bundles become isomorphic with each other. These are all the semistable <strong>sheaves</strong> with Chern classes c0, c1, c2. The geometric picture one gets from this is that the moduli space <strong>of</strong> semistable <strong>sheaves</strong> with these Chern classes consists <strong>of</strong> a double cover <strong>of</strong> the space <strong>of</strong> quadrics, branched over the discriminant locus. The space <strong>of</strong> quadrics in question is P 2 (we are talking about a net!), and the discriminant locus is a sextic curve in this P 2 . Therefore the moduli space is isomorphic to a double cover <strong>of</strong> P 2 branched over a sextic, which is again a K3 surface. Definition 5.1.8. Under the assumptions <strong>of</strong> the previous theorem, let M = M(v), and let α ∈ Br(M) be the obstruction <strong>of</strong> Definition 3.3.3 for this moduli problem. We’ll denote α by Obs(X, v). Theorem 3.3.2 shows the existence <strong>of</strong> a π ∗ M α-<strong>twisted</strong> universal sheaf U on X × M (where πM : X × M → M is the projection). Let A be an Azumaya algebra on M representing α, and using the equivalence between <strong>twisted</strong> <strong>sheaves</strong> and modules over an Azumaya algebra, represent U as a sheaf Ũ <strong>of</strong> modules over π∗ MA . Then Ũ is called a quasi-universal sheaf. It has the property that Ũ |X×[F] ∼ = F ⊕n for some n that only depends on Ũ (the isomorphism is as <strong>sheaves</strong> <strong>of</strong> OX-modules). This n is called the similitude <strong>of</strong> Ũ , denoted by s(Ũ ). (Here, F is a semistable sheaf on X with Mukai vector v, and [F ] is the point <strong>of</strong> M that corresponds to it.) In fact, Ũ satisfies a certain universal property which is very similar to that enjoyed by a universal sheaf; see [32, Appendix 2]. Definition 5.1.9. The cohomological Fourier-Mukai transform associated to Ũ is the homomorphism given by where and Ũ ∨ ϕ = ϕX→M : ˜ H(X, Q) → ˜ H(M, Q) ϕ( · ) = 1 s( Ũ )πM,∗(π ∗ X( · ).v( Ũ ∨ )), Ũ ∨ = RHom( Ũ , OX×M), v( Ũ ∨ ) = ch( Ũ ∨ ). td(X × M). 66
Theorem 5.1.10. Under the hypotheses <strong>of</strong> Theorem 5.1.6, the map ϕ is a Hodge isometry between ˜ H(X, Q) and ˜ H(M, Q). It maps v ∈ ˜ H(X, Q) to the vector (0, 0, ω) ∈ ˜ H(M, Q), and it therefore induces a Hodge isometry v ⊥ /v ∼ = H 2 (M, Q), the latter being computed inside ˜ H(X, Q). Restricted to v ⊥ /v, this isometry is independent <strong>of</strong> the choice <strong>of</strong> quasi-universal bundle and is integral, i.e. it takes integral vectors to integral vectors. It therefore induces a Hodge isometry the latter now computed in ˜ H(X, Z). H 2 (M, Z) ∼ = v ⊥ /v, Remark 5.1.11. Using the Torelli theorem, this gives a complete description <strong>of</strong> the moduli space M. Pro<strong>of</strong>. When the moduli problem is fine, one first proves that U (the universal sheaf) induces an equivalence <strong>of</strong> <strong>derived</strong> <strong>categories</strong> Db coh (X) ∼ = Db coh (M). We’ll do this in a more general context, in Theorem 5.5.1. Then the result follows immediately from the results in Section 3.1. The general case (when the moduli problem is not fine) is done by a deformation argument in [25, Chapter 6]. Theorem 5.1.12. Let n be the greatest common divisor <strong>of</strong> the numbers (u, v), where u runs over all ˜ H 1,1 (X) ∩ ˜ H(X, Z). Then the following statements hold: 1. There exist α-<strong>twisted</strong> locally free <strong>sheaves</strong> on M <strong>of</strong> ranks r1, r2, . . . , rk, with gcd(r1, r2, . . . , rk) = n, and therefore α is n-torsion. (α is the obstruction described in Definition 5.1.8.) 2. Denote by ϕ any cohomological Fourier-Mukai transform defined above (for some choice <strong>of</strong> quasi-universal sheaf). Then ϕ maps TX into TM, (viewing TX as a sublattice <strong>of</strong> ˜ H(X, Z) via the inclusion λ ↦→ (0, λ, 0)), and ϕ|TX is independent <strong>of</strong> the choice <strong>of</strong> quasi-universal bundle used to define it. 3. There exists λ ∈ TX such that v + λ is divisible by n (in ˜ H(X, Z)); for such a λ we have ϕ(λ) divisible by n (in TM). 4. ϕ is injective, and its cokernel is a finite, cyclic group <strong>of</strong> order n, generated by ϕ(λ)/n for any λ satisfying the condition in (3). Example 5.1.13. In Example 5.1.7 the number n = 2. Therefore the obstruction α (which lives in the Brauer group <strong>of</strong> the moduli space, which is the double cover <strong>of</strong> P 2 branched over a sextic) is <strong>of</strong> order 2. 67
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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: endowed with the product ((r, l, s)
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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