derived categories of twisted sheaves on calabi-yau manifolds

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

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

100 We start our study <strong>of</strong> the Jacobian by examining what happens locally. We are working with a generic elliptic fibration f : X → S, with a small open set U ⊆ S, and with the restriction f : XU → U <strong>of</strong> X to U. We assume U to be small enough so that f has a section s : U → XU. (See Theorem 6.1.8.) Proposition 6.4.2. For every point Q ∈ XU, let XQ be the fiber <strong>of</strong> f containing Q (Xf(Q)) and let P = s(f(Q)) be the corresponding point <strong>of</strong> the section in XQ. Then there exists a sheaf U on XU ×U XU, flat over the second factor, such that the fiber <strong>of</strong> U on XQ is isomorphic to O(P ) ⊗ IQ. Pro<strong>of</strong>. Note that since s is a section, P is always a smooth point <strong>of</strong> the fiber XQ. The image <strong>of</strong> the section s is a Cartier divisor on XU and as such defines a line bundle O(s) on XU. Take U = I∆ ⊗ O(s), where I∆ is the ideal sheaf <strong>of</strong> the diagonal ∆ in XU ×U XU. Using Lemma 6.3.7, it is easy to see that this sheaf satisfies the property we’re looking for. It is this sheaf U that will provide the local model for the α-<strong>twisted</strong> pseudouniversal sheaf whose existence is asserted in the introduction to this chapter. Note that if C is a reducible fiber <strong>of</strong> f, the section s picks up a distinguished component <strong>of</strong> C; changing the section will correspond to moving from l1 to l2 in Remark 6.3.10, and thus will correspond globally to performing a flop on the small resolution <strong>of</strong> the moduli space. Theorem 6.4.3. The family U <strong>of</strong> Proposition 6.4.2 (which depends on the choice <strong>of</strong> the section s) induces by the universal property <strong>of</strong> JU/U a surjective map XU → JU <strong>of</strong> schemes over U which is the contraction <strong>of</strong> those components <strong>of</strong> the fibers that do not meet the section s. Pro<strong>of</strong>. The sheaf U constructed before is semistable in each fiber, hence gives by the universal property <strong>of</strong> MU/U a map XU → MU <strong>of</strong> schemes over U. The image <strong>of</strong> this map is irreducible, and contains each [OXp] for all p ∈ S, so the image is in fact contained inside JU. Also, the image is closed (because X/S is proper) and three-dimensional (the map is locally injective around a point corresponding to a stable line bundle on a fiber <strong>of</strong> XU/U), which equals dim J. We conclude that the map XU → JU is surjective, and it is locally an isomorphism around the stable points <strong>of</strong> JU. Using the information from Theorem 6.3.11, and looking fiberwise at this map, we see that it contracts precisely the components <strong>of</strong> the fibers that are not hit by the section s (Remark 6.3.10). Theorem 6.4.4. Consider the full generic elliptic Calabi-Yau threefold X → S, and let J/S be its relative Jacobian. Then J/S is an elliptic fibration, and the only singularities <strong>of</strong> J are isolated ordinary double points (ODP’s), one over each I2 degeneration.
101 Pro<strong>of</strong>. Since X/S admits a local section (in the analytic topology), everything follows from Theorem 6.4.3 except the fact that the singularities <strong>of</strong> J are ODP’s. But this becomes obvious once we remark that each component <strong>of</strong> a reducible fiber has normal bundle O(−1) ⊕ O(−1) (Theorem 6.1.9), and it is well known that the contraction <strong>of</strong> such a curve in a threefold gives an ODP. Remark 6.4.5. There are a number <strong>of</strong> things to note here, and they are all related to the fact that J has singularities. One, is to note that the reason the singularities occur is the fact that the moduli space under consideration has properly semistable points, and these are precisely the singular points <strong>of</strong> J. Obviously, there is no hope in finding a universal sheaf on all <strong>of</strong> X ×S J (even allowing twistings), because there is no “good” candidate to put over the singular points <strong>of</strong> J (since they represent a whole S-equivalence class <strong>of</strong> <strong>sheaves</strong>). Also, it is worthwhile noting that it is not possible to have an equivalence Db coh (X) ∼ = Db coh (J, α) for some twisting α; indeed, this is because “<strong>derived</strong> <strong>categories</strong> detect smoothness” (this should be considered more as a slogan than a theorem), and D b coh (X) is “smooth” while Db coh (J, α) is not. We’ll see in the next section how to remedy this situation. The solution is to replace J by a small resolution <strong>of</strong> it ¯ J, by working in the analytic category. Finally, it is worthwhile noting that we cannot hope to find a small resolution <strong>of</strong> the singularities <strong>of</strong> J in the algebraic category. In order for a small resolution <strong>of</strong> J to exist, one would have to have rk Cl(J) > rk Pic(J) (we need to blow-up a Weil divisor which is not even Q-Cartier). One can now prove that rk Cl(J) = rk Cl(X) = rk Pic(X). Thus if one takes an X with rk Pic(X) = 2 (for example any <strong>of</strong> the examples in Section 6.2), then J cannot have a small resolution because we clearly have rk Pic(J) ≥ 2 (there is the section, which is a Cartier divisor, as well as divisors coming from Pic(S)). Theorem 6.4.6. For any analytic small resolution ρ : ¯ J → J <strong>of</strong> the singularities <strong>of</strong> J, there exists a covering {Ui} <strong>of</strong> S by analytic open sets such that for each i, are isomorphic, as analytic spaces over Ui. XUi and ¯ JUi Pro<strong>of</strong>. Let T be the collection <strong>of</strong> points in S over which in X there are degenerations <strong>of</strong> type I2. Cover S with analytic open sets {Ui} small enough to have the following properties: - for each i the map XUi → Ui has a section si; - each Ui contains at most one point <strong>of</strong> T . Let Xi = XUi , Ji = JUi and ¯ Ji = ¯ JUi . If Ui does not contain any points in T , the statement is clear: Xi ∼ = Ji ∼ = ¯ Ji since in this case Ji is smooth. Now assume Ui contains t ∈ T . The family U on Xi×Ui Xi that we constructed in Proposition 6.4.2 gives, by the universal property <strong>of</strong> Ji/Ui, a map ϕi : Xi → Ji that is a small resolution <strong>of</strong> Ji. Therefore Xi is isomorphic, over Ui, to one <strong>of</strong> the two small resolutions <strong>of</strong> J at x. If ¯ Ji happens to be that particular resolution, we’re done. Otherwise, replace the section si by another section s ′ i that
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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
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The reason for the notation is that
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Proof. The first s
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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 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: Having fixed a smooth point P ∈ C
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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