derived categories of twisted sheaves on calabi-yau manifolds

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

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

All we need to do now is show that we can find an F as above for which (v(F ), v) = 1 mod n. Indeed, we can then take u = v(F ) (using the fact the we have the freedom <strong>of</strong> choosing u as long as (u, v) = 1 mod n) and get U ∨ ϕX1→M1 (u)2 U ∨ = ϕX1→M1 (v(F ))2 = (v(F ), v) rk(V1) c1(V1) + integral part = (u, v) rk(V1) c1(V1) + integral part, which is what we want. Since we assume that n = 1 at t = 1 we can find a line bundle F on X1 with c1(F ) = d such that gcd(v0, v2.d, v4) = 1 (where v = (v0, v2, v4)). Since n divides v0 and v4, we must have gcd(v2.d, n) = 1. By possibly replacing F by some tensor power <strong>of</strong> itself, we can assume (v2.d) = 1 mod n. Then we have v(F ).v = v2.d − v4 − v0 − v0(d.d) = 1 mod n, as desired. Twisting F by OX1(−k) for k large enough, we get the F we were looking for, thus finishing the pro<strong>of</strong>. (Note that twisting by O(−k) does not change (v(F ), v) mod n, because (c1(OX1(1)).v2) must be divisible by n, since it equals (c1(OX0(1)).v2).) 5.4 The Map on Brauer Groups We have seen in the previous section that we can identify the obstruction α solely in terms <strong>of</strong> the Fourier-Mukai transform. In this section we prove that there is in fact a map between the Brauer groups <strong>of</strong> X and <strong>of</strong> M, whose kernel is generated by α. Lemma 5.4.1. Let M be a K3 surface, and let TM be the transcendental lattice <strong>of</strong> M, i.e. the orthogonal lattice to NS(M) inside H 2 (M, Z) (endowed with the intersection pairing). Then there is a natural isomorphism where T ∨ M is the dual lattice to TM. Br(M) = T ∨ M ⊗Z Q/Z, Pro<strong>of</strong>. Consider the exact sequence <strong>of</strong> Proposition 1.1.3, 0 → NS(M) ⊗ Q/Z → H 2 (M, Q/Z) → Br(M) → 0, (note that for a K3 M we have NS(M) = Pic(M)); using the universal coefficient theorem, we also know that H 2 (M, Q/Z) = H 2 (M, Z) ⊗ Q/Z, 78
and therefore Now consider the map given by Br(M) = (H 2 (M, Z)/ NS(M)) ⊗ Q/Z. p : H 2 (M, Z) → T ∨ M p(v)(w) = (v, w) for w ∈ TM. It is obviously a linear map, it is surjective (because H 2 (M, Z) is unimodular) and its kernel consists <strong>of</strong> all v ∈ H 2 (M, Z) such that (v, w) = 0 for all w ∈ TM, which is T ⊥ M . Since NS(M) is a primitive sublattice in H2 (M, Z), we have T ⊥ M = (NS(M) ⊥ ) ⊥ = NS(M), and therefore we conclude that thus obtaining the result <strong>of</strong> the lemma. T ∨ M = H 2 (M, Z)/ NS(M), Lemma 5.4.2. Let H be a unimodular lattice, let N be a primitive sublattice in H, and let n be a fixed integer. Define Nn to be the sublattice <strong>of</strong> N consisting <strong>of</strong> all x ∈ N such that (x, y) is divisible by n for all y ∈ N. Also, let T = N ⊥ , and let Tn be defined just as Nn. Then there is a natural group isomorphism Nn/nN ∼ = Tn/nT defined by sending an element v ∈ Nn/nN to the unique element λ <strong>of</strong> Tn/nT such that v − λ is divisible by n inside H. Pro<strong>of</strong>. Begin by proving that for every v ∈ Nn there exists a λ ∈ Tn such that v − λ is divisible by n. The idea is from [32]: consider the functional 1 n v∨ = 1 ∨ (v, ·) ∈ N n (it takes integer values because v ∈ Nn). Since N is a primitive sublattice in the unimodular lattice H, there exists w ∈ H such that w∨ |N = 1 nv∨ . Thus (v − nw, x) = 0 for all x ∈ N, and thus λ = v − nw ∈ T . Obviously, we have v − λ divisible by n inside H (equaling nw). The fact that λ is in fact in Tn is immediate: (λ, y) = (λ − v, y) = −n(w, y) for any y ∈ T . It is obvious that the map is well-defined, bijective, and a group homomorphism, so we’re set. 79
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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
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: for some k ≫ 0. By the same argum
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,
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derived categories of twisted sheaves on calabi-yau manifolds

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