derived categories of twisted sheaves on calabi-yau manifolds

More documents

Recommendations

Info

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

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

Chapter 3 Fourier-Mukai Transforms This chapter collects various results regarding integral functors and Fourier-Mukai transforms, as well as general results on the existence <strong>of</strong> relative moduli spaces and quasi-universal <strong>sheaves</strong>. The first section introduces integral functors between <strong>derived</strong> <strong>categories</strong> and their associated functors in cohomology, and examines the relationship between them. We draw our inspiration from [25, Chapter 6], although the case <strong>of</strong> odd dimensional cohomology classes is completely new. In the second section we sketch a pro<strong>of</strong> <strong>of</strong> the generalization (Theorem 3.2.1), to the case <strong>of</strong> <strong>twisted</strong> <strong>sheaves</strong>, <strong>of</strong> the criterion for when an integral functor is an equivalence. The original form <strong>of</strong> this criterion was obtained in a restricted form by Mukai, and in its general form by Bondal-Orlov and Bridgeland. We follow the general structure <strong>of</strong> Bridgeland’s excellent paper [5] with modifications for <strong>twisted</strong> <strong>sheaves</strong>. The chapter concludes with a section on the existence <strong>of</strong> relative moduli spaces and quasi-universal <strong>sheaves</strong>. Most <strong>of</strong> the statements here are known, and they are only included for future reference. 3.1 Integral Functors A fundamentally important class <strong>of</strong> morphisms between <strong>derived</strong> <strong>categories</strong> is the class <strong>of</strong> integral functors, introduced by Mukai. In this section we define them for the case <strong>of</strong> <strong>twisted</strong> <strong>derived</strong> <strong>categories</strong>. When the twisting is trivial, we also define the associated maps on cohomology, which leads to the definition <strong>of</strong> the Mukai intersection pairing and to the study <strong>of</strong> the relationship between the transforms on <strong>derived</strong> <strong>categories</strong> and the ones on cohomology. Definition 3.1.1. Let X and Y be proper and smooth schemes over C or compact 38
complex manifolds, and let X × Y πY ✲ Y πX ❄ X be the two projections. Let α ∈ ˇ H2 (Y, O∗ Y ), let ¯α = π∗ Y α and let U · ∈ Db coh (X × Y, ¯α −1 ). Then a functor <strong>of</strong> the form U · ΦY →X : D b coh(Y, α) → D b coh(X) U · L · ΦY →X( · ) = RπX,∗(U ⊗ Lπ ∗ Y ( · )) is called an integral functor. If it is an equivalence <strong>of</strong> <strong>categories</strong>, then it is called a Fourier-Mukai transform. In section 3.2 we’ll give a general criterion for when an integral functor is an equivalence (Theorem 3.2.1); however, for the rest <strong>of</strong> this section we’ll only be concerned with the case when X and Y are complex projective manifolds, and α = 0, so that the functor Φ U · Y →X takes Dbcoh (Y ) to Dbcoh 39 (X). For the remainder <strong>of</strong> this section we assume this to be the case. For the following definition it is necessary to observe that because we have td0(X) = 1 for any complex manifold X, we can express td(X) and ( td(X)) −1 by means <strong>of</strong> power series. Definition 3.1.2. Let and define by the formula H ∗ (X, C) = H i (X, C), i v : D b coh(X) → H ∗ (X, C) v(E · ) = ch(E · ). td(X). (The product here is the cup product in the cohomology <strong>of</strong> X. The exponential Chern character <strong>of</strong> a complex E · is defined to be ch(E · ) = (−1) i ch(E i ). It is easy to see that this definition descends to give a map i ch : D b coh(X) → H ∗ (X, C).)
Page 1 and 2: DERIVED CATEGORIES OF TWISTED SHEAV
Page 3 and 4: DERIVED CATEGORIES OF TWISTED SHEAV
Page 5 and 6: Tuturor celor din care am fost făc
Page 7 and 8: Table of Contents
Page 9 and 10: List of Figures 6.
Page 11 and 12: Introduction and Overview Twisted <
Page 13 and 14: twisted by α ∈
Page 15 and 16: fibers introduces interesting featu
Page 17 and 18: Chapter 1 Twisted Sheaves In this c
Page 19 and 20: 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: 2.4 Duality for Proper Smooth Morph
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 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,
show all

derived categories of twisted sheaves on calabi-yau manifolds

Create successful ePaper yourself

Delete template?

Save as template?