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 2 Derived Categories <strong>of</strong> Twisted Sheaves In this chapter we study the <strong>derived</strong> category <strong>of</strong> <strong>twisted</strong> <strong>sheaves</strong> on a scheme or analytic space, <strong>derived</strong> functors, and relationships among them. The theorems and pro<strong>of</strong>s here are quite technical, and could be skipped on a first reading, the general idea being that all the results that hold in the un<strong>twisted</strong> case also hold in the <strong>twisted</strong> case, with minor modifications. 2.1 Preliminary Results We start with a few remarks regarding injective and flat resolutions, and finiteness properties <strong>of</strong> these resolutions. Throughout this section X will denote a noetherian, separated scheme or analytic space, and α, α ′ , etc. are elements <strong>of</strong> H 2 (X, O ∗ X ). Lemma 2.1.1. Mod(X, α) has enough injectives for any α ∈ H 2 (X, O ∗ X ). Pro<strong>of</strong>. The pro<strong>of</strong> is the same as the one in [22, III, 2.2], using the correct f∗. Lemma 2.1.2. Any α-sheaf is the quotient <strong>of</strong> an OX-flat α-sheaf. Pro<strong>of</strong>. Let F be an α-sheaf on X. If U is any open set <strong>of</strong> X small enough to have α|U trivial, then OX,U (defined to be j!(OX|U), where j : U → X is the inclusion and j! is the one defined in Proposition 1.2.10) is an OX-flat α-sheaf, and any direct sum <strong>of</strong> such is flat. For every pair (U, f) where U is a small enough open set to have α|U trivial, and f is a section <strong>of</strong> F over U, consider the map OX,U → F that over U takes the constant section “one” to f (such a map can be found using the adjunction property <strong>of</strong> j! and j ∗ from Proposition 1.2.13), and take the direct sum <strong>of</strong> all these maps over all pairs (U, f). This is the desired surjection. Remark 2.1.3. The α-sheaf that surjects onto F , as constructed in the pro<strong>of</strong>, has the property that the stalk at each point <strong>of</strong> each underlying local sheaf is a free module over the local ring <strong>of</strong> the structure sheaf at that point. We’ll call this 26
property “free on stalks” for the purpose <strong>of</strong> this chapter. Note that these <strong>sheaves</strong> are not locally free, as they are not quasi-coherent! Lemma 2.1.4. If α ∈ Br(X), and if on X every coherent sheaf is the quotient <strong>of</strong> a locally free sheaf <strong>of</strong> finite rank (lffr), then the same holds for coherent α-<strong>sheaves</strong>. Pro<strong>of</strong>. Let G be an α −1 -lffr, and let F be any coherent α-sheaf; then F ⊗ G is a coherent sheaf on X, hence we can find a lffr L that surjects onto F ⊗ G . Tensor this surjection with G ∨ , to get a surjective map L ⊗G ∨ → F ⊗G ⊗G ∨ . But there is a surjective trace map G ⊗ G ∨ → OX given by the evaluation, and therefore a surjective composite map which is what we wanted. L ⊗ G ∨ → F ⊗ G ⊗ G ∨ → F Lemma 2.1.5. For any injective α-sheaf I and for any open set U ⊆ X we have I |U is an injective α-sheaf on U. In particular, if I = ({Ii}, {ϕij}) along an open cover U = {Ui}, then the sheaf Ii is injective on Ui for all i. Pro<strong>of</strong>. For the first part, use a pro<strong>of</strong> identical to [22, III, 6.1] (note that, in fact, that pro<strong>of</strong> can be written entirely in functorial terms, using the adjointness properties <strong>of</strong> f! and f ∗ , and using only the fact that f! takes injective maps to injective maps). For the last statement use Lemma 1.2.3 and the fact that the injectivity <strong>of</strong> an object is a categorical property. Proposition 2.1.6. Assume that X is smooth, <strong>of</strong> dimension n. Then for any coherent α-sheaf F and any coherent α ′ -sheaf G we have Ext i X(F , G ) = 0 for any i > n. (We define Ext i (F , · ) as the right <strong>derived</strong> functors <strong>of</strong> Hom(F , · ).) Remark 2.1.7. Note that we are, unfortunately, unable to conclude from this proposition that any coherent α-sheaf has injective dimension at most n on X. Indeed, to prove this, we would need to prove that the property “Ext i X(F , I ) = 0 for any coherent F and any i > 0” for an α-sheaf I implies that I is injective. Of course, if I were coherent, this property would say that I is indeed injective in Coh(X, α). But we can not hope, in general, to find resolutions <strong>of</strong> coherent α-<strong>sheaves</strong> by coherent injective α-<strong>sheaves</strong>, so we are forced to go to the larger category <strong>of</strong> non-coherent α-<strong>sheaves</strong>. On a scheme, a possible alternative would be to use the results in [23, II.7], but we do not know <strong>of</strong> similar results for analytic spaces. Fortunately, we can get by with the slightly weaker form <strong>of</strong> Proposition 2.1.6. 27
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: Definition 1.3.18. Two R-algebras A
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 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
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In order to identify the kernel, no
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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
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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
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113 One space of i
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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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