derived categories of twisted sheaves on calabi-yau manifolds

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

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

so that the H 3,3 component <strong>of</strong> ch(U · ). td(X × Y ) is just ch3(U · ). Indeed, π ∗ X c2(X) ∈ H 4,0 , so it can not give an element <strong>of</strong> H 3,3 by multiplication with anything. We have ch3(U · ) = 1 6 (c3 1(U · ) − 3c1(U · )c2(U · ) + 3c3(U · )); 110 c1(U · ) ∈ H 2 (X × Y, C), and since H 1 (X, C) = H 1 (Y, C) = 0, it must belong to H 2,0 ⊕ H 0,2 . Therefore c 3 1(U · ) can not have any H 3,3 component. Similarly for c1(U · )c2(U · ). Hence we conclude that the only contribution to the map H 3 (X, C) → H 3 (Y, C) comes from c3(U · ), in the form and hence the result. ϕ| H 3 (X,C)( · ) = 1 2 πY,∗(π ∗ X( · ).c ′ 3(U · )), Proposition 6.7.2. Just as before, let X and Y be Calabi-Yau threefolds with equivalent <strong>derived</strong> <strong>categories</strong>. Assume that there exists a codimension 2 subvariety i : Z ↩→ X × Y which is locally a complete intersection, and an element U · ∈ Db coh (Z) such that if one takes i∗U · ∈ Db · coh (X × Y ), then Φi∗U X→Y : Dbcoh (X) → (Y ) is an equivalence <strong>of</strong> <strong>categories</strong>. Let D b coh KZ = c1(ωZ) ∈ H 2 (X × Y, Z) be the first Chern class <strong>of</strong> the dualizing sheaf <strong>of</strong> Z, (which exists since Z is locally a complete intersection). If KZ is divisible by 2 in H 2 (X ×Y, Z), then the isometry <strong>of</strong> Proposition 6.7.1 is integral, i.e. it restricts to a Hodge isometry H 3 (X, Z)free ∼ = H 3 (Y, Z)free. Pro<strong>of</strong>. As seen before, to prove the integrality <strong>of</strong> the isomorphism we would need to show that c3(i∗U · ) is divisible by 2. We use Grothendieck-Riemann-Roch to compute c3(i∗U · ) (we can apply it because i is a locally complete intersection projective morphism). We have ch(i∗U · ) = i∗(ch(U · ). td(−N )), where N is the normal sheaf <strong>of</strong> Z in X × Y , and the negation is taken in the Grothendieck group (see [21, Exposé 0]). The left hand side <strong>of</strong> the above equality is r + c1 + 1 2 (c2 1 − c2) + 1 6 (c3 1 − 3c1c2 + 3c3) + . . . ,
111 where ci = ci(i∗U · ). Obviously, r = c1 = 0 because Z has complex codimension 2 (also from the right hand side <strong>of</strong> the equality), so the real dimension 6 part <strong>of</strong> the left hand side consists <strong>of</strong> just 1 2 c3(i∗U · ). On the other hand, the real dimension 6 part <strong>of</strong> the right hand side comes from the dimension 2 part <strong>of</strong> ch(U · ). td(−N ), which consists <strong>of</strong> We conclude that c1(U · ) − 1 2 rk(U · )c1(N ). c3(i∗U · ) = 2i∗c1(U · ) − rk(U · )i∗c1(N ). The first term is obviously divisible by 2. Using the adjunction formula and the fact that X and Y have trivial canonical bundles, we have so the result follows. KZ = c1(ωZ) = c1(N ), Theorem 6.7.3. Let X and Y be the elliptic Calabi-Yau threefolds considered in Theorem 6.6.2, assuming that the base S has canonical class divisible by 2 (for example, if S = P 1 ×P 1 ). Then the Fourier-Mukai transform given by the universal sheaf induces a Hodge isometry H 3 (X, Z)free ∼ = H 3 (Y, Z)free. Pro<strong>of</strong>. Obviously the support <strong>of</strong> the universal sheaf U is X ×S Y , which is <strong>of</strong> codimension 2 in X × Y . It is also a locally complete intersection, because it is the pull-back <strong>of</strong> the diagonal in S × S (which is a locally complete intersection) under the natural map X × Y → S × S. Therefore the conditions <strong>of</strong> the previous theorem apply, and we only need to show that KX×SY is divisible by 2. We have the commutative square X ×S Y πY ✲ Y πX pY ❄ pX ❄ X ✲ S Since all the spaces involved are Cohen-Macaulay, we have equalities (in the corresponding K-groups, or integral cohomology groups) KX = p ∗ XKS + KX/S KY = p ∗ Y KS + KY/S KX×SY = π ∗ XKX + KX×SY/X.
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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
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Part II Applications 49
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X which would intersect a general f
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to C ′ , independent of</
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Jij → S. This means that if Fij w
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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
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: 109 where the intersection form on
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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