derived categories of twisted sheaves on calabi-yau manifolds

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

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

and p(F ; t) < t ⇔ p ′ (F ; t) < t, Therefore, if F is a semistable sheaf on C with p(F ; t) = t, then F is also semistable with respect to the second polarization, and the same statement holds with “semistable” replaced by “stable” instead. Pro<strong>of</strong>. There are two possibilities: dim F is 0 or 1. The first case is trivial: p(F ; t) = p ′ (F ; t) = 1. Assume dim F = 1, so that P (F ; t) = a1t + χ(F ), P ′ (F ; t) = a ′ 1t + χ(F ), with a1, a ′ 1 > 0. Since p(F ; t) < t, χ(F ) < 0, so that p ′ (F ; t) < t as well. If F is semistable with p(F ; t) = t, then clearly p ′ (F ; t) = t, and any destabilizing sheaf with respect to the second polarization would be destabilizing with respect to the first one as well. On C there is a special class <strong>of</strong> <strong>sheaves</strong>, which is described below. Lemma 6.3.4. For any two distinct points P, Q ∈ C, or for P = Q smooth points <strong>of</strong> C, there exists a unique non-trivial extension 0 → IQ → F (P, Q) → OP → 0, which is precisely OC(P ) ⊗ IQ if P is smooth. (Note that P is a Cartier divisor on C in this case, so it makes sense to talk about OC(P ), which is a line bundle on C.) Pro<strong>of</strong>. All we need to do is compute Ext 1 (OP , IQ) and show it is equal to C. A trivial local computation shows that Ext i OP if i = 1 (OP , IQ) = 0 otherwise. Since the cohomology H i (C, OP ) vanishes except when i = 0, the local-to-global spectral sequence for Ext’s gives the claim. Proposition 6.3.5. Any extension F = F (P, Q) given by the previous lemma is semistable with Hilbert polynomial P (F ; t) = 2t. Such a sheaf is stable if and only if P and Q are smooth points <strong>of</strong> C belonging to the same component <strong>of</strong> C. If it is semistable but not stable, its Jordan-Hölder filtration has factors Ol1(−1) and Ol2(−1) (and hence all semistable <strong>sheaves</strong> among the F (P, Q)’s are in the same S-equivalence class). Pro<strong>of</strong>. The exact sequence 0 → IQ → OC → OQ → 0 together with P (OC; t) = 2t shows that P (IQ; t) = 2t − 1, and therefore the Hilbert polynomial P (F ; t) = 2t. Let G be a dimension zero subsheaf <strong>of</strong> F , and let H be the kernel <strong>of</strong> the map G → OP . It is a subsheaf <strong>of</strong> IQ, and has dimension zero, so it must be zero because IQ is pure. Therefore G is a subsheaf <strong>of</strong> OP , so it could only be OP . 94
But this is impossible, because then F would be the trivial extension OP ⊕ IQ, contradiction. We conclude that F has no zero-dimensional sub<strong>sheaves</strong>, hence it is pure. Let H be a subsheaf <strong>of</strong> IQ. Viewing H as a subsheaf <strong>of</strong> OC via the natural inclusion IQ → OC, we can consider the closed subscheme H that is determined by H . Note that H is a subscheme <strong>of</strong> C and hence it is reduced at the generic point <strong>of</strong> each component <strong>of</strong> C. H could be one <strong>of</strong> the following: 1. Supp H = C; in this case H is a nilpotent ideal sheaf, hence 0 because C is reduced. We have P (H ; t) = 0. 2. H is supported on one component, say l1, and possibly at some other isolated points on l2; then P (OH; t) = t + 1 + k where k ≥ 0 takes into account the extra isolated points on l2, as well as the possible nonreducedness <strong>of</strong> H at the singular points <strong>of</strong> C. (Recall that we have P (O P 1; t) = t+1 by Riemann- Roch.) Therefore P (H ; t) = t − k − 1. Since Q ∈ H, if Q is on l2 − l1 then necessarily k ≥ 1. 3. H is supported at a number <strong>of</strong> points; then P (H , t) = 2t−k for some k ≥ 1, and if H = IQ then k ≥ 2. Let G be a nonzero proper subsheaf <strong>of</strong> F , and consider the composite map G → F → OP . There are two cases to consider, when this map is zero or when it is surjective. In the first case, G is a subsheaf <strong>of</strong> IQ so by the previous analysis p(G ; t) = t − k/n where n is 1 or 2 and k ≥ 1 and hence p(G ; t) < t = p(F ; t), so that G cannot be a destabilizing sheaf for F . Assume we’re in the second case, and consider H to be the kernel <strong>of</strong> the composite map G → F → OP , which is a subsheaf <strong>of</strong> IQ. Since we have the exact sequence 0 → H → G → OP → 0 we see that we have P (G ; t) = P (H ; t) + 1. Checking each possibility for H in turn, we see that indeed p(G ) ≤ t = p(F ). Since we proved that p(G ; t) ≤ p(F ; t) for all proper sub<strong>sheaves</strong> G <strong>of</strong> F , we conclude that F is semistable. Now assume P and Q are lying in the same component <strong>of</strong> C, and none is a singular point <strong>of</strong> C. Without loss <strong>of</strong> generality, assume P, Q ∈ l1. In order to have p(G ; t) = t for some proper subsheaf G <strong>of</strong> F , the map G → OP must be surjective, and P (H ; t) = t−1 (H is, as before, the kernel <strong>of</strong> G → OP ). (If P (H ; t) = 2t−1, then H = IQ, and hence G = F , contradicting our assumption that G is a proper subsheaf <strong>of</strong> F .) Therefore H = Il1 or H = Il2. Since H ⊆ IQ, and Q ∈ l2, H must be Il1. But then H is zero around P , hence G is the trivial extension between H and OP . This is a contradiction, because if this were the case OP would be a subsheaf <strong>of</strong> F , contradicting the fact that F is pure. We conclude that p(G ; t) < t for all proper sub<strong>sheaves</strong> G <strong>of</strong> F . 95
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DERIVED CATEGORIES OF TWISTED SHEAV
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DERIVED CATEGORIES OF TWISTED SHEAV
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Tuturor celor din care am fost făc
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Table of Contents
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List of Figures 6.
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Introduction and Overview Twisted <
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twisted by α ∈
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fibers introduces interesting featu
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Chapter 1 Twisted Sheaves In this c
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to be H2 an(X, O∗ X ). If we do n
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is bijective. Then A is called an A
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such that α = δa and β = δb. Si
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Lemma 1.2.3. Let α ∈ Č2 (X, O
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Proof. Let U ′
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Proof. Since E and
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Proof. Follows fro
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Definition 1.3.12. Let A be a ring.
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Definition 1.3.18. Two R-algebras A
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property “free on stalks” for t
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2.2 The Derived Category and Derive
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to reduce to the case when F · als
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So assume n > 0, and as before cons
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Proposition 2.3.4. Let f : X → Y
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2.4 Duality for Proper Smooth Morph
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complex manifolds, and let X × Y
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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: Remark 6.3.2. This theorem basicall
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,
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derived categories of twisted sheaves on calabi-yau manifolds

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