derived categories of twisted sheaves on calabi-yau manifolds

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

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

and Br(J) are isomorphic, as abstract groups. Indeed, they are isomorphic to T ∨ X ⊗ Q/Z and T ∨ J ⊗ Q/Z, respectively, and TX is isomorphic to TJ as abstract groups. However, if one takes into account the lattice structure on TX and TJ, one obtains the more precise formulation <strong>of</strong> Theorem 5.4.3. Remark 5.4.6. Identifying Br(M) with T ∨ M ⊗ Q/Z, we can view any α ∈ Br(M) as a functional on TM, with values in Q/Z. Theorem 5.4.3 states that TX is Hodge isometric (via the Fourier-Mukai transform) to Ker(α), viewed as a Hodge sublattice <strong>of</strong> TM. The striking thing about this is the following: if one takes αk (we keep the multiplicative notation for the operation in Br(M)) for gcd(k, ord(α)) = 1, then obviously Ker(αk ) = Ker(α). Thus, if we have X and X ′ K3 surfaces, and v ∈ ˜ H1,1 (X, Z), v ′ ∈ ˜ H1,1 (X ′ , Z) such that MX(v) = M = MX ′(v′ ), and Obs(M, v) = α, Obs(M ′ , v ′ ) = αk , then TX is Hodge isometric to TX ′. This will be used in the following section. 5.5 Relationship to Derived Categories Whenever one deals with a moduli problem <strong>of</strong> semistable <strong>sheaves</strong> on a K3 X, and the moduli space M is 2-dimensional and has no properly semistable points, then one gets an equivalence <strong>of</strong> <strong>derived</strong> <strong>categories</strong> between some <strong>derived</strong> category <strong>of</strong> <strong>twisted</strong> <strong>sheaves</strong> on M, and the <strong>derived</strong> category <strong>of</strong> (usual) coherent <strong>sheaves</strong> on X. This is the content <strong>of</strong> the following theorem. Theorem 5.5.1. Let X be a K3 surface, let v ∈ ˜ H 1,1 (X, Z) be a primitive, isotropic Mukai vector, and assume that there is a polarization on X such that the moduli space M = MX(v) <strong>of</strong> semistable <strong>sheaves</strong> with Mukai vector v is nonempty and does not contain any properly semistable points. Let α = Obs(X, v). Then we have D b coh(X) ∼ = D b coh(M, α −1 ). Pro<strong>of</strong>. Using Theorem 3.3.2 we conclude that there exists a π∗ Mα-<strong>twisted</strong> universal sheaf U on X × M. Therefore we can consider the integral functor F = Φ U M→X : D b coh(M, α −1 ) → D b coh(X) defined by U , and in order to check that F is an equivalence, all we have to do is check that the conditions <strong>of</strong> Theorem 3.2.1 are satisfied. For a point [F ] ∈ M that corresponds to a stable sheaf F on X with Mukai vector v, we have F (O[F]) = F , from the very definition <strong>of</strong> a universal sheaf. Thus the conditions <strong>of</strong> Theorem 3.2.1 can be restated as: 1. if F = G , Ext i (F , G ) = 0 for all i; 2. Hom(F , F ) = C, and Ext i (F , F ) = 0 for all i < 0 or i > 2; 3. F ⊗ ωX ∼ = F ; 82
for all F , G stable <strong>sheaves</strong> on X with Mukai vector v. Since X is smooth <strong>of</strong> dimension 2, Ext i (F , G ) = 0 for i < 0 or i > 2 anyway. Since F and G are stable, Hom(F , G ) = 0 if F = G . Therefore Ext 2 (F , G ) = 0 by Serre duality (use the fact that ωX is trivial). Since χ(F , G ) = (v(F ), v(G )) = (v, v) = 0 by Lemma 3.1.7, we conclude that Ext 1 (F , G ) = 0 as well. Hom(F , F ) = C because a stable sheaf is simple. Finally, the last condition holds automatically because ωX is trivial. As an application <strong>of</strong> this we prove the following: Theorem 5.5.2. Let M be a K3 surface, let α ∈ Br(M), and let k ∈ Z be such that gcd(k, ord(α)) = 1. Assume that there exist K3 surfaces X and X ′ , and Mukai vectors v ∈ ˜ H 1,1 (X, Z), v ′ ∈ ˜ H 1,1 (X ′ , Z) such that we have isomorphisms M ∼ = MX(v) ∼ = MX ′(v′ ), and such that under these isomorphisms, Obs(X, v) −1 = α and Obs(X ′ , v ′ ) −1 = α k . Then we have D b coh(M, α) ∼ = D b coh(M, α k ). Pro<strong>of</strong>. We have seen (Remark 5.4.6) that under these assumptions TX is Hodge isometric to TX ′. Orlov’s theorem (Theorem 5.1.14) then tells us that D b coh(X) ∼ = D b coh(X ′ ). But Theorem 5.5.1 shows that D b coh (M, α) ∼ = D b coh (X) and Db coh (M, αk ) ∼ = D b coh (X′ ), and thus D b coh(M, α) ∼ = D b coh(M, α k ). It is tempting (especially in view <strong>of</strong> the results <strong>of</strong> Chapter 6) to conjecture that the conclusion <strong>of</strong> the previous theorem holds more generally: Conjecture 5.5.3. For any K3 surface M, and any twisting α ∈ Br(M), we have for any k with gcd(k, ord(α)) = 1. D b coh(M, α) ∼ = D b coh(M, α k ) The difficulty in proving such a theorem would lie on one hand in the fact that if one takes TX = Ker(α), then TX may not be embeddable in the K3 lattice LK3. Even if one assumes this embeddability condition, there are problems with finding the Mukai vector v. 83
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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
Page 27 and 28:
Proof. Let U ′
Page 29 and 30:
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
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 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: In order to identify the kernel, no
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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