derived categories of twisted sheaves on calabi-yau manifolds

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

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

Chapter 5 K3 surfaces In this chapter we apply the general results deduced in Part I to different situations related to K3 surfaces. The reason K3 surfaces provide an interesting set <strong>of</strong> examples is because, on one hand, we have a good explicit description <strong>of</strong> their moduli space (the Torelli theorem, 5.1.1) so we can handle them easily in terms <strong>of</strong> lattices and Hodge structures, and, on the other hand, moduli spaces <strong>of</strong> <strong>sheaves</strong> on K3’s are well understood, primarily through the work <strong>of</strong> Mukai ([32]) (see the introductory section 5.1). Furthermore, it is also known when two K3 surfaces have equivalent <strong>derived</strong> <strong>categories</strong> (<strong>of</strong> un<strong>twisted</strong> <strong>sheaves</strong>), by the work <strong>of</strong> Orlov ([36]). The first section presents the known results about K3 surfaces, moduli spaces <strong>of</strong> <strong>sheaves</strong> on them, and <strong>derived</strong> <strong>categories</strong> <strong>of</strong> un<strong>twisted</strong> <strong>sheaves</strong>. Everything else is built upon these facts. We have seen in Chapter 4 that one can recover the classical Ogg-Shafarevich theory <strong>of</strong> elliptic fibrations without a section by studying the obstruction to the existence <strong>of</strong> universal <strong>sheaves</strong> for certain moduli problems. For K3 surfaces, we can do better: in most cases we don’t need an elliptic fibration on the K3 to play the same game. In Section 2 we study deformations <strong>of</strong> vector bundles as <strong>twisted</strong> <strong>sheaves</strong>, obtaining results that will be used in identifying the obstruction. These results are also interesting on their own, in the context <strong>of</strong> deformation theory <strong>of</strong> vector bundles. Section 3 contains the main technical result <strong>of</strong> this chapter, an explicit description <strong>of</strong> the obstruction α (whose existence is asserted in Theorem 3.3.2) to the existence <strong>of</strong> a universal sheaf for an arbitrary moduli problem on a K3 X, when the moduli space is again a K3. This description is given in purely cohomological terms, via Fourier-Mukai transforms. By analogy with Ogg-Shafarevich theory, one can also attempt to reverse the process <strong>of</strong> going from the pair (X, v) (where X is a K3, and v is the data for a moduli problem on X) to the pair (M, α) (where M = M(v) is the moduli space, and α is the obstruction to the existence <strong>of</strong> a universal sheaf on X × M). This is what would be needed in order to obtain a complete generalization <strong>of</strong> Ogg-Shafarevich theory in terms only <strong>of</strong> moduli problems and obstructions. Unfortunately, we cannot push this too far. The best we can do is to find the transcendental lattice <strong>of</strong> the candidate X, together with its Hodge structure. In most cases, this should 62
allow us to get X; however, finding v seems a more difficult problem. The last section deals with the relationship <strong>of</strong> the results <strong>of</strong> the previous sections with the topic <strong>of</strong> equivalences <strong>of</strong> <strong>derived</strong> <strong>categories</strong>. This provides an insight into why we can only get a hold <strong>of</strong> the transcendental lattice <strong>of</strong> X in generalizing Ogg- Shafarevich theory. As a consequence <strong>of</strong> this analysis we also observe the following curious phenomenon: when we can find X and v, X only depends on the cyclic subgroup <strong>of</strong> Br(M) generated by α. This allows us to infer that in some cases one has D b coh(M, α) = D b coh(M, α k ) for (k, ord(α)) = 1, a rather striking application <strong>of</strong> the theoretical results <strong>of</strong> the previous chapters. In the next chapter we’ll see a similar result for elliptic Calabi- Yau threefolds. 5.1 General Facts The main results about K3 surfaces: the Torelli theorem, Mukai’s calculation <strong>of</strong> moduli spaces <strong>of</strong> <strong>sheaves</strong> on K3’s, Orlov’s characterization <strong>of</strong> <strong>derived</strong> equivalences for K3’s – are all presented in this section, and put in the context <strong>of</strong> the results in the first part. The Torelli Theorem Let’s fix notation: let X be a complex K3 surface, i.e., a complex manifold <strong>of</strong> complex dimension 2, having trivial canonical class (KX = 0) and being simply connected (equivalent, using Hodge theory, to having H 1 (X, OX) = 0). We have H 2 (X, Z) = Z 22 , and considering this group with the intersection pairing we obtain a lattice which is isomorphic to LK3 = (E8) ⊕2 ⊕ (U(1)) ⊕3 . Inside the H 2 (X, Z) lattice there are two natural sublattices, the Néron-Severi sublattice <strong>of</strong> X, NS(X), (consisting <strong>of</strong> first Chern classes <strong>of</strong> holomorphic vector bundles), and its orthogonal complement, the transcendental lattice TX = NS(X) ⊥ . Both these lattices are primitive sublattices <strong>of</strong> H 2 (X, Z), but may be non-unimodular. The complex structure <strong>of</strong> X is reflected in its Hodge decomposition H 2 (X, C) = H 2,0 (X) ⊕ H 1,1 (X) ⊕ H 0,2 (X). We have dim H 2,0 (X) = 1, so in order to specify this decomposition, it is enough to specify a one-dimensional complex subspace π ⊆ H 2 (X, C) (called the period <strong>of</strong> X): indeed, one can take H 2,0 (X) to be π, H 0,2 to be the complex conjugate π <strong>of</strong> π, and H 1,1 to be the orthogonal (with respect to the intersection pairing) to the span <strong>of</strong> π and π. The complex subspace π satisfies the following extra property: if v ∈ π is a non-zero vector, we have v.v = 0 and v.v > 0. 63
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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
Page 21 and 22: 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: Remark 4.5.3. Note that we have fix
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 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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