derived categories of twisted sheaves on calabi-yau manifolds

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

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

Chapter 4 Smooth Elliptic Fibrations In this chapter we consider the first non-trivial example <strong>of</strong> an occurrence <strong>of</strong> <strong>twisted</strong> <strong>sheaves</strong>, in the construction <strong>of</strong> the <strong>twisted</strong> Poincaré bundle for an elliptic fibration without a section. Here we only analyze the case <strong>of</strong> fibrations without singular fibers, and investigate the relationship <strong>of</strong> our approach to Ogg-Shafarevich theory. Most <strong>of</strong> the interesting phenomena will only occur when we’ll take into account fibrations with singular fibers, which we’ll do in Chapter 6, but this chapter should be viewed as a “birational analysis” <strong>of</strong> what comes later. Also, the results in this chapter serve to illustrate some <strong>of</strong> the concepts introduced in the previous chapters. 4.1 Elliptic Fibrations Definition 4.1.1. A smooth, projective morphism <strong>of</strong> smooth schemes (or analytic spaces), pX : X → S, such that for all s ∈ S, Xs (the fiber <strong>of</strong> pX over s) is a curve <strong>of</strong> genus one over k(s), is called a smooth elliptic fibration. Note that although Xs has lots <strong>of</strong> points defined over k(s) when s is a closed point <strong>of</strong> S, Xs might not have any point defined over k(s) when s is, for example, η, the generic point <strong>of</strong> S. This will be the main issue in the study <strong>of</strong> elliptic fibrations. In fact, most <strong>of</strong> the results that will be discussed in this chapter are concerned primarily with what happens around η, so that in many cases we’ll be able to remove the loci in S where “bad” things happen, and still get our results. Just in order to have a specific example in mind, we give here an example <strong>of</strong> a smooth elliptic fibration. Example 4.1.2. Let X be a general bidegree (3, 3) hypersurface in P 2 × P 2 , and consider the map f : X → P 2 given by projection on the first factor <strong>of</strong> the product P 2 ×P 2 . Using Bertini’s theorem X is smooth and the general fiber <strong>of</strong> f is smooth. If ∆ ⊆ P 2 is the discriminant locus <strong>of</strong> this fibration (the locus over which the fiber is not smooth), and if S = X \ ∆, then XS = X × P 2 S → S is a smooth elliptic fibration. In this example we can in fact see that f does not have a rational section. Indeed, the closure <strong>of</strong> a rational section over all <strong>of</strong> P 2 would be a divisor D in 50
X which would intersect a general fiber <strong>of</strong> the map X → P 2 in one point. But Lefschetz’s theorem tells us that the map Pic(P 2 × P 2 ) → Pic(X) is surjective. Since all divisors in Pic(P 2 × P 2 ) have intersection number with the general fiber <strong>of</strong> X → P 2 divisible by 3, we conclude that we cannot find a rational section in XS → S. On the other hand, the pull-back <strong>of</strong> the general hyperplane section <strong>of</strong> the second factor <strong>of</strong> P 2 × P 2 provides an example <strong>of</strong> a 3-section (a flat, effective divisor that meets each fiber in 3 points). Therefore we conclude that Xη has no rational points (a rational point would correspond to a rational section <strong>of</strong> XS → S), but that it has a rational divisor <strong>of</strong> degree 3 (corresponding to the 3-section). 4.2 The Relative Jacobian Given a smooth elliptic fibration pX : X → S, there is a naturally constructed elliptic fibration pJ : J → S that has the same fibers as X → S over the closed points <strong>of</strong> S, but which has a section. This fibration, called the relative Jacobian <strong>of</strong> X → S is in a certain sense a universal object, and it will constitute the starting point <strong>of</strong> our study <strong>of</strong> smooth elliptic fibrations. Definition 4.2.1. Fix a relatively ample sheaf OX(1) for the morphism pX : X → S, and let pJ : J → S be the relative moduli space <strong>of</strong> semistable <strong>sheaves</strong> <strong>of</strong> rank 1, degree 0 on the fibers <strong>of</strong> pX (see Theorem 3.3.1 for existence and properties). The fibration J → S (which is a smooth elliptic fibration) is called the relative Jacobian <strong>of</strong> X → S. Proposition 4.2.2. The map pJ is a smooth elliptic fibration with a section. Over a closed point s ∈ S, we have Js ∼ = Xs. If we set L = pJ,∗ωJ/S, then any smooth elliptic fibration p ′ : J ′ → S possessing a section, having J ′ s ∼ = Xs for all closed points s ∈ S, and for which p ′ ∗ωJ ′ /S ∼ = L , is isomorphic (over S) to J → S. Pro<strong>of</strong>. The map pJ is a flat, projective morphism by the very definition <strong>of</strong> relative moduli spaces. The flat family OX <strong>of</strong> <strong>sheaves</strong> <strong>of</strong> rank 1, degree 0 on the fibers <strong>of</strong> pX gives by the universal property <strong>of</strong> J → S a map s0 : S → J which is a section <strong>of</strong> pJ. The fiber Js <strong>of</strong> pJ over a closed point s ∈ S is isomorphic to Pic ◦ (Xs), and since Xs is a smooth elliptic curve over an algebraically closed field, we conclude that Js ∼ = Xs. The smoothness <strong>of</strong> J follows at once from local dimension estimates (see [25, Section 4.5]). Therefore pJ is a smooth elliptic fibration. Now we can apply [13, 2.2 and 2.3] to J ′ → S to obtain the uniqueness property <strong>of</strong> pJ. Note that since all the fibers <strong>of</strong> pX are connected, the same holds for pJ, and therefore the birational morphism g that is given to us by [loc.cit.] is in fact an isomorphism. A final note: since locally (in the étale or analytic topologies) pX has sections (because there are no multiple fibers), this pro<strong>of</strong> actually shows that X is locally isomorphic to J, over S. We’ll give a direct pro<strong>of</strong> <strong>of</strong> this fact in the next proposition. 51
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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
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 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: Part II Applications 49
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
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103 Therefore, α can be represente
Page 115 and 116:
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
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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