derived categories of twisted sheaves on calabi-yau manifolds

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

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

116 2. If one has Db coh (X, α) ∼ = Db coh (Y ) for smooth spaces X and Y , and for α ∈ Br ′ (X), does it imply that α ∈ Br(X)? This would have implications for the conjecture that Br(X) = Br ′ (X) for a smooth space X. 3. What are the conditions for being able to find a smooth space Y with Db coh (X, α) ∼ = Db coh (Y ), given X and α? 4. What is the most general setup in which one has D b coh(X, α) ∼ = D b coh(X, α k ) for smooth X, α ∈ Br(X), k coprime to the order <strong>of</strong> α? 5. Is there more geometry to the “virtual” K3 surfaces from Remark 5.5.4? 6. In Example 6.2.3, is X 2 deformation equivalent to X? This would give a complete counterexample to the Torelli problem (and it would also prove that X and Y are not birational to each other). Note that in order to prove this, it would be enough to solve the following problem: let C be a smooth curve <strong>of</strong> genus 1, let r and d be coprime positive integers, and let V be an irreducible vector bundle <strong>of</strong> rank r and degree d. Then one would need to find a natural isomorphism H 0 (C, V ) ∼ = H 0 (C, det V ). 7. What kind <strong>of</strong> singularities can arise through processes similar to the one that generate the singularities <strong>of</strong> the relative Jacobian? In all the “generic” examples I know <strong>of</strong> these singularities are ordinary double points. Are there other cases, more degenerate? 8. How should one understand the Vafa-Witten, Aspinwall-Morrison example? What is the relationship <strong>of</strong> <strong>twisted</strong> <strong>sheaves</strong> with physics? The main cause we cannot solve more <strong>of</strong> these issues is our lack <strong>of</strong> examples. These would likely provide us with a better understanding <strong>of</strong> the theory <strong>of</strong> <strong>twisted</strong> <strong>sheaves</strong>, which would bring forth answers to some <strong>of</strong> these questions.
Bibliography [1] Aspinwall, P., Morrison, D., Gross, M., Stable singularities in string theory, Commun. Math. Phys. 178 (1996), 135-146 [2] Bănică, C., Stănă¸silă, O., Algebraic Methods in the Global Theory <strong>of</strong> Complex Spaces, John Wiley, New York (1976) [3] Barth, W., Peters, C., Van de Ven, A., Compact Complex Surfaces, Erg. Math., 3. Folge, Band 4, Springer Verlag, Berlin (1984) [4] Bondal, A., Orlov, D., Semiorthogonal decompositions for algebraic varieties, preprint, alg-geom/9506012 [5] Bridgeland, T., Equivalences <strong>of</strong> triangulated <strong>categories</strong> and Fourier-Mukai transforms, preprint, alg-geom/9809114 [6] Bridgeland, T., Fourier-Mukai transforms for elliptic surfaces, J. reine angew. math. 498 (1998) 115-133 (also preprint, alg-geom/9705002) [7] Bridgeland, T., Maciocia, A., Fourier-Mukai transforms for K3 fibrations, preprint, alg-geom/9908022 [8] Bridgeland, T., King, A., Reid, M., Mukai implies McKay, preprint, alg-geom/9908027 [9] Bucur, I., Deleanu, A., Introduction to the Theory <strong>of</strong> Categories and Functors, John Wiley, New York (1968) [10] Chatterjee, D. S., On the Construction <strong>of</strong> abelian gerbs, Ph.D. thesis, Cambridge (1998) [11] Cossec, F., Dolgachev, I., Enriques Surfaces I, Birkhäuser, Boston, (1989) [12] Deleanu, A., Jurchescu, M., Andreian-Cazacu C., Topologie, Categorii, Suprafet¸e Riemanniene, Ed. Acad. RSR, Bucharest (1966) [13] Dolgachev, I., Gross, M., Elliptic three-folds I: Ogg-Shafarevich theory, J. Alg. Geom. 3 (1994), 39-80 (also preprint, alg-geom/9210009) 117
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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
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It is easy to see that ϕi(ti) = [O
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Remark 4.5.3. Note that we have fix
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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 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: Open Questions and Further Directio
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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