derived categories of twisted sheaves on calabi-yau manifolds

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

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

group <strong>of</strong> this algebra is isomorphic to PGL(n) by the Skolem-Noether theorem, we get a map from the set <strong>of</strong> Azumaya algebras <strong>of</strong> rank n 2 to H 1 (X, PGL(n)). In fact, it is easy to see that the set <strong>of</strong> isomorphism classes <strong>of</strong> Azumaya algebras <strong>of</strong> rank n 2 coincides with the set H 1 (X, PGL(n)), and that the equivalence relation on Azumaya algebras that we introduced does nothing more than quotient out the image <strong>of</strong> the map H 1 (X, GL(n)) → H 1 (X, PGL(n)). We quote here, without pro<strong>of</strong>, some results about Brauer groups. Theorem 1.1.9. Br(X) is torsion. Pro<strong>of</strong>. Obvious from the previous theorem. Theorem 1.1.10. If X is a smooth curve, Br(X) = Br ′ (X) = 0. If X is a smooth surface, then Br(X) = Br ′ (X). Pro<strong>of</strong>. The first statement follows easily from the exponential exact sequence. For the second one, see [30, IV.2.16]. Theorem 1.1.11. If X is a proper scheme over C, and if h : X h → X is the natural continuous map from the associated analytic space X h , then the map A ↦→ A h = h ∗ (A ) takes an Azumaya algebra A on X to an Azumaya algebra A h on X h and induces an isomorphism <strong>of</strong> Br(X) with Br(X h ). Pro<strong>of</strong>. Use [20, Exposé XII, 4.4] and [30, IV.2.1c] to construct an inverse. Elements <strong>of</strong> H2 (X, O∗ X ) as Gerbes There is a third way to describe the elements <strong>of</strong> H2 (X, O∗ X ), similar to the way one describes line bundles via transition functions. The ideas (and the term gerbe) come from Giraud’s non-abelian cohomology (see [16]), <strong>of</strong> which we’ll only use a tiny amount. I follow roughly the ideas in [24], where much more detail can be found. The idea is that we are comfortable thinking <strong>of</strong> objects that are defined on intersections <strong>of</strong> two open sets in a cover (like the transition functions <strong>of</strong> a line bundle), but we feel uneasy thinking <strong>of</strong> things defined on triple intersections. Thus we would prefer a description <strong>of</strong> the objects in terms <strong>of</strong> tw<strong>of</strong>old intersections. ) will be called gerbes. Any gerbe will be locally trivial Elements <strong>of</strong> H2 (X, O∗ X (although nobody tells us what a trivial gerbe is, unlike the case <strong>of</strong> a line bundle, for which we have a clear geometric description). For example, if α ∈ H2 (X, O∗ X ) is given by a Čech cocycle {αijk} on the threefold intersections <strong>of</strong> an open cover {Ui}, then the gerbe will be trivial along the open sets in the cover. Now we would like to understand what the “transition functions” are, i.e. what is the difference between two locally isomorphic gerbes. One way to understand this is via Čech cohomology: let α and β be trivial cocycles on the open sets U and V <strong>of</strong> X, that agree on U ∩ V . This means that there are 1-cochains a and b 12
such that α = δa and β = δb. Since α = β on U ∩ V , this means that δ(a − b) = 0 on U ∩ V , and therefore a − b is a 1-cocycle on U ∩ V . It thus represents a line bundle L on U ∩ V (recall that all cochains, cocycles, etc. take values in O ∗ X ). We conclude that one can give the following description <strong>of</strong> gerbes: a gerbe is given by fixing an open cover {Ui}, and giving a collection {Lij} on the tw<strong>of</strong>old intersections Ui∩Uj, under the requirement that this collection satisfies the following cocycle conditions: 1. Lii = OUi ; 2. Lij = L −1 ji ; 3. Lij ⊗ Ljk ⊗ Lki =: Lijk is trivial (but a trivialization is not being fixed); 4. Lijk ⊗ L −1 jkl ⊗ Lkli ⊗ L −1 lij is canonically trivial (i.e. we have chosen an isomorphism <strong>of</strong> it with OUi∩Uj∩Uk∩Ul ). One way to understand conditions 3 and 4 above is by considering what happens when one refines the cover: one can get to a situation where all the line bundles Lij are trivial, hence all the information must lie in the actual bundles, and not in their isomorphism type. A choice <strong>of</strong> trivialization for Lijk would correspond then to a choice <strong>of</strong> an element <strong>of</strong> Γ(Ui ∩ Uj ∩ Uk, O∗ X ), and we return to the old description <strong>of</strong> gerbes via Čech cohomology, where condition 4 corresponds to the fact that we are dealing with a cocycle. These conditions will automatically be satisfied in all the constructions we’ll do. For an explicitly worked example, see Chapter 4. We would also like to express what it means to modify a gerbe (given by the above collection <strong>of</strong> line bundles) by a coboundary. Let {Fi} be line bundles on {Ui}. Then, modifying {Lij} by ∂{Fi} gives the collection {Lij ⊗ Fi ⊗ F −1 j } (and therefore {Lij} represents the trivial gerbe if and only if one can find line bundles {Fi} such that Lij = Fi ⊗ F −1 j ). For more details about the gerbe representation, the reader should consult [10] or the standard reference on non-abelian cohomology, [16]. 1.2 Twisted Sheaves If one considers Azumaya algebras (or, more generally, gerbes) as replacements for the structure sheaf <strong>of</strong> a scheme, then <strong>twisted</strong> <strong>sheaves</strong> are the natural objects to replace <strong>sheaves</strong> <strong>of</strong> modules. In this section we give their definition and basic properties, and show a first example where they naturally appear. For an extended example, see Chapter 4. Throughout this section, (X, OX) will be a scheme (considered with the étale topology) or an analytic space (with either the étale or analytic topology). 13
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: is bijective. Then A is called an A
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 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
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Proof. Trivial cha
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and therefore rk(V ) = (v(V ), (0,
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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
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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
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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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