derived categories of twisted sheaves on calabi-yau manifolds

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

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

elieve is true), it should not be isomorphic to X 2 , but at the moment we are unable to prove this. 6.3 Jacobians <strong>of</strong> Curves <strong>of</strong> Genus 1 Given a generic elliptic Calabi-Yau threefold X → S, we will be primarily interested in understanding its relative Jacobian, which is defined to 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> X → S. Therefore, it is fundamentally important to study the corresponding moduli problems on the curves that can occur as fibers <strong>of</strong> X → S. We do this in this section. One note about the definition we use for stability <strong>of</strong> <strong>sheaves</strong>. We use Gieseker stability, in its generalized form for pure <strong>sheaves</strong> used by Simpson in [41]. For a quick account <strong>of</strong> these notions, see [25, Section 1.2]. Theorem 6.3.1. Let C be an irreducible curve <strong>of</strong> arithmetic genus 1 (i.e. either <strong>of</strong> the following: a smooth elliptic curve, or a rational curve with one singular point which is either a node or a cusp), and let J be the moduli space <strong>of</strong> semistable <strong>sheaves</strong> on C <strong>of</strong> rank 1, degree 0. (Since C is irreducible, the rank does not depend on the choice <strong>of</strong> polarization.) Then we have: 1. J is isomorphic to C; 2. there are no properly semistable points in J; 3. OC is a stable sheaf <strong>of</strong> rank 1, degree 0, and is therefore represented by a point [OC]; 4. choose any isomorphism ϕ : C → J, let Γ ⊆ C × J be the graph <strong>of</strong> ϕ, and let P = ϕ −1 ([OC]), which is a smooth point <strong>of</strong> C. Since P is a Cartier divisor on C, it makes sense to speak <strong>of</strong> the line bundle OC(P ) on C. Then U = IΓ ⊗ π ∗ C(OC(P )) is a universal sheaf on C × J, where IΓ denotes the ideal sheaf <strong>of</strong> Γ. Any smooth point <strong>of</strong> C can occur as P . Pro<strong>of</strong>. The result is well-known (see [22, II, 6.10.1, 6.11.4 and Ex. 6.7]). The only fine point is, in the case <strong>of</strong> one <strong>of</strong> the singular curves, the fact that the compactification J <strong>of</strong> CaCl ◦ (C) is given by the addition <strong>of</strong> one extra point, which corresponds to the stable sheaf OC(−P ) ⊗ O(Q), where Q is the singular point and O(Q) is defined to be the dual <strong>of</strong> IQ. But this is also standard (the pro<strong>of</strong> would be entirely similar to the one given in the analysis <strong>of</strong> the moduli problem on an I2 curve, given in the sequel). 92
Remark 6.3.2. This theorem basically tells us that, as long as we do not have reducible fibers, nothing very interesting happens: the relative moduli space will only consist <strong>of</strong> some rearrangement <strong>of</strong> the fibers <strong>of</strong> the original space. Note that the usual way to parametrize line bundles <strong>of</strong> degree zero on an elliptic curve (or the other singular curves that the theorem deals with) is to associate to a point Q ∈ C the line bundle O(Q − P ), where P is the origin. Our construction associates O(P − Q) to Q ∈ C, an entirely isomorphic construction. We chose this particular one for ease <strong>of</strong> notation later on. (When Q is singular, it is easy to refer to O(−Q) as IQ, but there is no easy way to refer to its dual.) Note that this convention is different from the one used in Chapter 4. From here on we concentrate on understanding what is a corresponding result to Theorem 6.3.1, for a curve <strong>of</strong> type I2, i.e. a curve which consists <strong>of</strong> two components, l1 and l2, each isomorphic to P 1 , and meeting transversely at two points. Fix such a curve C for the rest <strong>of</strong> the section. On C fix the polarization whose restriction to each component P 1 is O P 1(1). We’ll study <strong>sheaves</strong> F on C which are semistable (with respect to this polarization) and have Hilbert polynomial P (F ; t) = 2t (note that this is the Hilbert polynomial <strong>of</strong> OC by a simple calculation). This corresponds to studying semistable <strong>sheaves</strong> <strong>of</strong> rank 1 and degree 0, under the correct definition <strong>of</strong> rank (see, for example, [25, 1.2.2]). This may seem like an arbitrary choice <strong>of</strong> polarization, but in fact the next lemma shows that the notion <strong>of</strong> (semi)stability does not depend on the choice <strong>of</strong> polarization, and therefore this particular choice is just a convenience. Recall the definition <strong>of</strong> the reduced Hilbert polynomial: the (usual) Hilbert polynomial <strong>of</strong> a sheaf F <strong>of</strong> dimension n (the dimension <strong>of</strong> a sheaf is the dimension <strong>of</strong> its support) is always <strong>of</strong> the form P (F ; t) = n ait i with an > 0, and the reduced Hilbert polynomial <strong>of</strong> F is defined to be i=0 p(F ; t) = 1 P (F ; t). an Reduced Hilbert polynomials are compared lexicographically: p(F ; t) ≤ p(G ; t) if and only if the leading coefficient <strong>of</strong> p(G ; t) − p(F ; t) is non-negative. Lemma 6.3.3. Consider C polarized by two polarizations; for a coherent sheaf F on C, write P (F ; t) and p(F ; t) for the usual and the reduced Hilbert polynomials, respectively, with respect to the first polarization, and P ′ (F ; t) and p ′ (F ; t) with respect to the second one. Then we have, for any coherent sheaf F on C, p(F ; t) ≤ t ⇔ p ′ (F ; t) ≤ t, 93
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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
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 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: contract 45 curves 2H+D-E contract
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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