derived categories of twisted sheaves on calabi-yau manifolds

More documents

Recommendations

Info

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

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

Definition 1.2.1. Let α ∈ Č2 (X, O∗ X ) be a Čech 2-cocycle (in the topology under consideration), given by means <strong>of</strong> an open cover U = {Ui}i∈I and sections αijk ∈ ). We define an α-<strong>twisted</strong> sheaf on X (or α-sheaf, in short) to Γ(Ui ∩ Uj ∩ Uk, O∗ X consist <strong>of</strong> a pair ({Fi}i∈I, {ϕij}i,j∈I), with Fi being a sheaf <strong>of</strong> OX-modules on Ui and ϕij : Fj|Ui∩Uj → Fi|Ui∩Uj being isomorphisms such that 1. ϕii is the identity for all i ∈ I; 2. ϕij = ϕ −1 ji for all i, j ∈ I; 3. ϕij ◦ ϕjk ◦ ϕki is multiplication by αijk on Fi|Ui∩Uj∩Uk for all i, j, k ∈ I. A homomorphism f between α-<strong>twisted</strong> <strong>sheaves</strong> F , G consists <strong>of</strong> a collection <strong>of</strong> maps fi : Fi → Gi for each i ∈ I such that fi ◦ ϕF,ij = ϕG ,ij ◦ fj for all i, j ∈ I. Composition <strong>of</strong> homomorphisms is defined in the obvious way, and this makes the class <strong>of</strong> all α-<strong>sheaves</strong> and homomorphisms (given along the cover U) into a category Mod(X, α, U). We’ll see (Lemma 1.2.3) that for different open covers on which α can be represented we get equivalent <strong>categories</strong>, so we’ll just denote any <strong>of</strong> these equivalent <strong>categories</strong> by Mod(X, α). Also, Lemma 1.2.8 shows that for α and α ′ in the same cohomology class the <strong>categories</strong> Mod(X, α) and Mod(X, α ′ ) are equivalent, so we will use the notation Mod(X, α) for α ∈ ˇ H2 (X, O∗ X ) whenever the choice <strong>of</strong> the specific cocycle is irrelevant. Often we’ll even abuse the notation by saying “let F ∈ Mod(X, α), G ∈ Mod(X, α ′ ), etc. for some α, α ′ , . . . ∈ ˇ H2 (X, O∗ X ),” and mean by this “let α, α ′ , . . . be any cocycles in Č2 (X, O ∗ X 14 ), let U be an open cover over which α, α ′ , . . . can be represented, and let F be an α-sheaf, G an α ′ -sheaf, etc. given over U.” We will never deal with an infinite number <strong>of</strong> α’s at the same time, so there is no problem finding a cover that works for all simultaneously. An α-sheaf F on X is called coherent if all the underlying <strong>sheaves</strong> Fi are coherent. The category <strong>of</strong> α-<strong>twisted</strong> coherent <strong>sheaves</strong> will be denoted by Coh(X, α). One defines similarly the category <strong>of</strong> quasi-coherent α-<strong>sheaves</strong>, Qcoh(X, α). It is easy to see that Mod(X, α), Coh(X, α) and Qcoh(X, α) are abelian <strong>categories</strong>, under the natural definitions <strong>of</strong> kernel, cokernel, etc. Before we start into the general theory <strong>of</strong> <strong>twisted</strong> <strong>sheaves</strong>, let’s consider a typical example. It is a continuation <strong>of</strong> Example 1.1.1. For another, more extensive example, see Chapter 4. Example 1.2.2. Consider the setup <strong>of</strong> Example 1.1.1. Recall that we covered X by open sets Ui, and on each Ui we found a vector bundle Ei <strong>of</strong> rank n and isomorphisms ¯ϕij : Ej → Ei with ¯ϕij ◦ ¯ϕjk ◦ ¯ϕki = αijk · id . This collection forms an α-<strong>twisted</strong> sheaf E for the α defined by the collection {αijk} (which is easily seen to be a cocycle). Making different choices for the isomorphisms ¯ϕ only changes α by a coboundary. We’ll see in Section 1.3 that End(E ) is naturally an Azumaya algebra, and that E is a <strong>twisted</strong> sheaf <strong>of</strong> modules over this algebra.
Lemma 1.2.3. Let α ∈ Č2 (X, O ∗ X ), and let U′ be a refinement <strong>of</strong> an open cover U on which α can be represented. Then we have an equivalence <strong>of</strong> <strong>categories</strong> Mod(X, α, U) ∼ = Mod(X, α, U ′ ). Before we prove this result, it is useful to recall the following result on gluing <strong>sheaves</strong> in the étale topology. Lemma 1.2.4 (Gluing Sheaves in the Étale Topology). Let X be a scheme, endowed with the étale topology, let U = {ρi : Ui → X} be an open cover <strong>of</strong> X, and suppose we are given for each i a sheaf Fi on Ui, and for each i, j an isomorphism such that for each i, j, k, ϕij : (p ij j )∗ Fj|Ui×XUj → (pij i )∗ Fi|Ui×XUj (p ijk ij )∗ (ϕij) ◦ (p ijk jk )∗ (ϕjk) ◦ (p ijk ki )∗ (ϕki) = id (p ijk i ) ∗ Fi , where p ijk ij is the projection from Ui ×X UJ ×X Uk to Ui ×X Uj, and similarly for the other projections. Then there exists a unique sheaf F on X, together with isomorphisms ψi : ρ ∗ i F ∼ → Fi such that for each i, j, (p ij i )∗ (ψi) = ϕij ◦ (p ij j )∗ (ψj) on Ui ×X Uj. We say loosely that F is obtained by gluing the <strong>sheaves</strong> Fi via the isomorphisms ϕij. Remark 1.2.5. We have phrased this lemma using pull-backs instead <strong>of</strong> restriction maps in order to make apparent its relationship to the standard lemma in descent theory (see, for example, [30, I.2.22]). From here on, however, we’ll revert to the more convenient notation where if F is a sheaf on a space X, and ϕ : U → X is an étale open set, we write F |U for ϕ ∗ F , and if f : F → G is a map <strong>of</strong> <strong>sheaves</strong> on X, we write f|U for ϕ ∗ (f). See also the definition <strong>of</strong> a presheaf in [30, p.47]. Pro<strong>of</strong>. (We only sketch a pro<strong>of</strong>, since this is well-known. For a topological space, see [22, Ex. II.1.22].) Let U → X be an open set in the étale topology. Define F (U) = {(si) ∈ Fi(Ui ×X U) | ϕij(sj|Ui×XUj×XU) = si|Ui×XUj×XU for all i, j}, and define ψi by i ψi(U)((sj)j) = si ∈ Fi(Ui ×X U). It is now only a tedious check to see that F is a sheaf and that the collection {ψi} satisfies the required properties. The important thing to note, however, is that we have not made any choices here, and therefore this construction is entirely functorial. 15
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: such that α = δa and β = δb. Si
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 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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?