derived categories of twisted sheaves on calabi-yau manifolds

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

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

which is exact in both the étale and analytic topologies. The long exact cohomology sequence gives Pic(X) ·n −→ Pic(X) c1 −→ H 2 (X, Z/nZ) → Br ′ (X) ·n −→ Br ′ (X) which implies that the n-torsion part <strong>of</strong> Br ′ (X), Br ′ (X)n, fits in the exact sequence 0 → Pic(X) ⊗ Z/nZ → H 2 (X, Z/nZ) → Br ′ (X)n → 0. Taking the direct limit over all n, we get the result. We also have: Theorem 1.1.4. If X is a smooth scheme then, in the étale topology, Br ′ (X) is torsion. For the associated analytic space, X h , we have Br ′ ét(X) = Br ′ an(X h )tors. Pro<strong>of</strong>. The first statement is just [18, II, 1.4]. For the second statement note that H 2 (X, Q/Z) and Pic(X) are the same in the étale and analytic topologies, and use the previous theorem. There are two particular cases where we want to specialize these results further. One is the case <strong>of</strong> a smooth, simply connected surface. In this case we have H 3 (X, Z) = 0 and H 2 (X, Z) is torsion free and isomorphic to H2(X, Z) by Poincaré duality. From the universal coefficient theorem we get H 2 (X, Q/Z) ∼ = H 2 (X, Z) ⊗ Q/Z, so we conclude that Br ′ (X) = (H 2 (X, Z)/ NS(X)) ⊗ Q/Z. The other case <strong>of</strong> interest is when X is a Calabi-Yau manifold. In this case we have Pic(X) ∼ = H 2 (X, Z) because H 1 (X, OX) = H 2 (X, OX) = 0, so we conclude from the above two theorems and from the universal coefficient theorem that Br ′ (X) ∼ = H 3 (X, Z)tors. Azumaya Algebras and the Brauer Group Of particular importance in the study <strong>of</strong> <strong>twisted</strong> <strong>sheaves</strong> will be those elements <strong>of</strong> the cohomological Brauer group that arise as δ([Y ]) in Example 1.1.1, i.e. those that lie in the images <strong>of</strong> the maps H 1 (X, GL(n)) 10 δ −→ Br ′ (X) for various n. This is obviously a subgroup <strong>of</strong> Br ′ (X), which is called the Brauer group <strong>of</strong> X and denoted by Br(X). In what follows we’ll give a more intrinsic description <strong>of</strong> it. Definition 1.1.5. Let R be a commutative ring, and let A be a (non-commutative) R-algebra. Assume that A is finitely generated projective as an R-module, and that the canonical homomorphism A ⊗R A ◦ → EndR(A)
is bijective. Then A is called an Azumaya algebra over R. The sheaf <strong>of</strong> algebras A = Ã over Spec R is called a sheaf <strong>of</strong> Azumaya algebras over R. (We follow the notation <strong>of</strong> [22, Section II.5], using Ã for the sheafification <strong>of</strong> A.) If X is a scheme, A a sheaf <strong>of</strong> algebras over X, then A is said to be a sheaf <strong>of</strong> Azumaya algebras over X if and only if over each affine open set <strong>of</strong> the form Spec R, A is isomorphic to the sheafification <strong>of</strong> Ã <strong>of</strong> an Azumaya algebra A over R. It is an easy exercise to prove that this definition is consistent. (See, for example, [30, IV, 2.1].) The following theorem details the local structure <strong>of</strong> Azumaya algebras: Theorem 1.1.6. Let X be a scheme, A a sheaf <strong>of</strong> algebras on X which is locally free <strong>of</strong> finite rank as a sheaf <strong>of</strong> OX-modules. Then A is a sheaf <strong>of</strong> Azumaya algebras if and only if for every x ∈ X there exists a neighborhood (étale or analytic) U → X <strong>of</strong> x and a locally free sheaf E on U such that AU is isomorphic (as an OU-algebra) with EndOU (E ). Pro<strong>of</strong>. See [18, II, 5.1] for the étale case; the analytic case follows immediately from the étale one. The set <strong>of</strong> isomorphism classes <strong>of</strong> Azumaya algebras has a natural group structure under the tensor product operation, using the easy fact that End(E ) ⊗OU End(E ′ ) ∼ = End(E ⊗OU E ′ ) and the previous theorem. One can consider the equivalence relation given by setting two Azumaya algebras A and A ′ to be equivalent if and only if there exist global vector bundles E and E ′ on X such that A ⊗ End(E ) ∼ = A ′ ⊗ End(E ′ ). This is easily seen to be well defined and to be compatible with the group operation, hence one can take the quotient to obtain a new group. Definition 1.1.7. The group <strong>of</strong> isomorphism classes <strong>of</strong> Azumaya algebras on X, under the tensor product operation, modulo the equivalence relation defined above, is called the Brauer group <strong>of</strong> X and is denoted by Br(X). The relevance <strong>of</strong> Azumaya algebras in the context <strong>of</strong> the cohomological Brauer group stems from the following result: Theorem 1.1.8. There exists a natural injective map Br(X) → Br ′ (X), whose image coincides with the collection <strong>of</strong> all obstructions <strong>of</strong> the form δ([Y ]) (for various n) under the maps δ : H 1 (X, PGL(n)) → H 2 (X, O ∗ X ). Pro<strong>of</strong>. We’ll only sketch the pro<strong>of</strong>, since it is well known (see the standard references quoted in the beginning <strong>of</strong> this section). The way one constructs the map Br(X) → Br ′ (X) is by noting that since an Azumaya algebra <strong>of</strong> rank n 2 can be viewed as a bundle whose fibers are <strong>of</strong> the form Mn(OU), and the automorphism 11
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: to be H2 an(X, O∗ X ). If we do n
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 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
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allow us to get X; however, finding
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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
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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
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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
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Chapter 6 Elliptic Calabi-Yau Three
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Definition 6.1.2. A projective morp
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Example 6.2.1. We’ve already seen
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contract 45 curves 2H+D-E contract
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Remark 6.3.2. This theorem basicall
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But this is impossible, because the
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We have H 0 (C, IQ) = 0 and χ(IQ)
Page 109 and 110:
Having fixed a smooth point P ∈ C
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101 Proof. Since X
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103 Therefore, α can be represente
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Corollary 6.5.6. Under the hypothes
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107 Remark 6.6.4. We could also pro
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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
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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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