derived categories of twisted sheaves on calabi-yau manifolds

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

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

[14] Eisenbud, D., Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics Vol. 150, Springer-Verlag (1995) 118 [15] Eisenbud, D., Popescu, S., Walter, C., Enriques surfaces and other non- Pfaffian subcanonical subschemes <strong>of</strong> codimension 3, MSRI preprint 037, (1999) [16] Giraud, J., Cohomologie non-abélienne, Grundlehren Vol. 179, Springer- Verlag (1971) [17] Gross, M., Finiteness theorems for elliptic Calabi-Yau threefolds, Duke Math. J., Vol. 74, No. 2 (1994), 271-299 [18] Grothendieck, A., Le groupe de Brauer I–III, in Dix Exposés sur la Cohomologie des Schémas, North-Holland, Amsterdam (1968), 46-188 [19] Grothendieck, A., Dieudonné, J., Eléments de Géométrie Algébrique III: Étude cohomologique des faisceaux cohérents, Publ. Math. IHES, 11 (1961) and 17 (1963) [20] Grothendieck, A. et al., Séminaire de Géométrie Algébrique 1, Revêtements Étales et Groupe Fondemental, Lecture Notes in Math. Vol. 224, Springer- Verlag (1971) [21] Grothendieck, A., Berthelot, P., Illusie, L., et al, Séminaire de Géométrie Algébrique 6: Théorie des Intersections et Théorème de Riemann-Roch, Lecture Notes in Math. 225, Springer-Verlag, Heidelberg (1971) [22] Hartshorne, R., Algebraic Geometry, Graduate Texts in Mathematics Vol. 52, Springer-Verlag (1977) [23] Hartshorne, R., Residues and Duality, Lecture Notes in Mathematics Vol. 20, Springer-Verlag (1966) [24] Hitchin, N. J., Lectures on Special Lagrangian Submanifolds, Lectures given at the ICTP School on Differential Geometry, April 1999, preprint, math.DG/9907034 [25] Huybrechts, D., Lehn, M., Geometry <strong>of</strong> Moduli Spaces <strong>of</strong> Sheaves, Aspects in Mathematics Vol. E31, Vieweg (1997) [26] Kontsevich, M., Homological algebra <strong>of</strong> mirror symmetry, Proceedings <strong>of</strong> the 1994 International Congress <strong>of</strong> Mathematicians I, Birkäuser, Zürich, 1995, p. 120; alg-geom/9411018 [27] Lam, T. Y., Lectures on Modules and Rings, Graduate Texts in Mathematics Vol. 189, Springer-Verlag (1999)
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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
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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 and 126: Open Questions and Further Directio
Page 127: Bibliography [1] Aspinwall, P., Mor
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