derived categories of twisted sheaves on calabi-yau manifolds

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

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

Pro<strong>of</strong>. First, note that for any open set U ⊆ X we have Hom U(F |U, G |U) = Hom X(F , G )|U (where by restriction we mean the usual pull-back via the inclusion U ↩→ X). Also, for any injective α-sheaf I on X we have I |U injective (Lemma 2.1.5). So we can use the pro<strong>of</strong> <strong>of</strong> [22, III, 6.2] to conclude that we have Ext i U(F |U, G |U) ∼ = Ext i X(F , G )|U. This allows us to reduce the problem to the case when X is affine (or Stein) and the cover U = {Ui} on which we are working contains X, as the open set U0. Since F and G are coherent, we conclude that Ext i X(F0, G0) = 0 for i > n, where F0 and G0 are the <strong>sheaves</strong> on U0 in the representation <strong>of</strong> F and G . (Use the fact that for an affine scheme X = Spec A, and for coherent <strong>sheaves</strong> F = ˜ M and G = Ñ, we have ExtiX(F , G ) = Ext i A(M, N)˜, and that a regular ring <strong>of</strong> dimension n has global dimension n.) Since all the <strong>sheaves</strong> that make up the <strong>twisted</strong> sheaf Ext i X(F , G ) are isomorphic to restrictions <strong>of</strong> Ext i X(F0, G0) to smaller open sets, we conclude that they are all zero, hence Ext i X(F , G ) = 0. Proposition 2.1.8. If X is smooth, <strong>of</strong> dimension n, then any α-sheaf F has a finite flat resolution <strong>of</strong> length at most n. Pro<strong>of</strong>. Using Lemma 2.1.2, construct a flat resolution Gn ϕ −→ Gn−1 → · · · → G0 → F → 0 by α-<strong>sheaves</strong> that are free on stalks (see Remark 2.1.3). Over each open set in the cover U = {Ui} that we are working on we get a resolution <strong>of</strong> Fi by <strong>sheaves</strong> Gn,i → Gn−1,i → · · · → G0,i → Fi → 0, where each Gk,i is free on stalks. Considering the corresponding exact sequence on stalks, and using Lemma 19.2, Theorem 19.2 and Theorem 2.5 in [29], we conclude that the kernel <strong>of</strong> the map Gn,i → Gn−1,i is free on stalks. Therefore replacing the initial resolution by 0 → Ker ϕ → Gn−1 → · · · → G0 → F → 0 we obtain a resolution <strong>of</strong> F all <strong>of</strong> whose terms are α-<strong>sheaves</strong> that are free on stalks. This is the desired OX-flat resolution. Proposition 2.1.9. Let f : Y → X be a proper morphism <strong>of</strong> schemes or analytic spaces, whose fibers have dimension at most n. Let α ∈ ˇ H2 (X, O∗ X ), and let F be a coherent f ∗α-<strong>twisted</strong> sheaf on Y . Then Rif∗F is a coherent α-<strong>twisted</strong> sheaf for all i, and is zero for i > n. (We define Rif∗ in the usual way, as the right <strong>derived</strong> functors <strong>of</strong> the left exact functor f∗ : Mod(Y, f ∗α) → Mod(X, α).) Pro<strong>of</strong>. Using Corollary 1.2.6 represent F as ({Fi}, {ϕij}) along an open cover {f −1 (Ui)}, for some cover {Ui} <strong>of</strong> X. Using Lemma 2.1.5, it is easy to see that computing R i f∗F in the category <strong>of</strong> <strong>twisted</strong> <strong>sheaves</strong> (using <strong>twisted</strong> injective resolutions) gives the same result as gluing together R i f∗Fi along the isomorphisms R i f∗ϕij. Now the result follows from the corresponding results for un<strong>twisted</strong> <strong>sheaves</strong> on schemes or analytic spaces (see [19, 3.2.1] and [2]). 28
2.2 The Derived Category and Derived Functors Definition 2.2.1. The α-<strong>twisted</strong> <strong>derived</strong> category <strong>of</strong> coherent <strong>sheaves</strong>, denoted by Db coh (X, α), is the bounded <strong>derived</strong> category <strong>of</strong> the abelian category Mod(X, α), with all cohomology α-<strong>sheaves</strong> being coherent. In a similar fashion we define D + coh (X, α), Kbcoh (X, α), K+ coh (X, α), etc. Remark 2.2.2. For all the notations pertaining to the <strong>derived</strong> category, we use the notations set up in [23, I.4]. For <strong>derived</strong> functors, our reference is [23, I.5]. Remark 2.2.3. Let X · be a complex such that H i (X · ) = 0 for all i > n0 for some n0, and let Y · = · · · → X n0−1 → Ker d n0 → 0 → · · · . Then it is easy to see that there is a natural injective map Y · → X · which is a quasi-isomorphism. Similarly, if H i (X · ) = 0 for all i < n0, then X · there is a natural surjective quasi-isomorphism X · → Y · , where Y · = · · · → 0 → Coker d n0−1 → X n0+1 → · · · . Let D be the full subcategory <strong>of</strong> Dcoh(X, α) consisting <strong>of</strong> complexes whose cohomology is zero except inside a bounded range. There is a natural functor Db coh (X, α) → D, and an easy application <strong>of</strong> [23, I.3.3] and <strong>of</strong> the previous comments shows that this functor is an equivalence <strong>of</strong> <strong>categories</strong>. Therefore we’ll use the name “bounded complex” for either a complex which is zero outside a bounded range, or for one whose cohomology is zero outside a bounded range, and the context will make clear which one we mean (if it matters). Theorem 2.2.4. If X is a scheme or analytic space, α ∈ ˇ H2 (X, O∗ X ), and if f : Y → X is a morphism from another scheme or analytic space, then the following <strong>derived</strong> functors are defined: R Hom · : Dcoh(X, α) ◦ × D + coh (X, α) → Dcoh(Ab), RHom · : Dcoh(X, α) ◦ × D + coh (X, α′ ) → Dcoh(X, α −1 α ′ ), L ⊗ : D − coh (X, α) × D− coh (X, α′ ) → D − coh (X, αα′ ), Lf ∗ : D − coh If f is also proper then is also defined. (X, α) → D− coh (Y, f ∗ α). Rf∗ : Dcoh(Y, f ∗ α) → Dcoh(X, α) Pro<strong>of</strong>. All these functors are defined exactly as in [23, II.2-II.4], using Propositions 2.1.1 and 2.1.2 to ensure the existence <strong>of</strong> the respective <strong>derived</strong> functors. Note that we need properness <strong>of</strong> f for Rf∗ in order to insure that the cohomology <strong>sheaves</strong> are coherent. 29
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: property “free on stalks” for t
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 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
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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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