derived categories of twisted sheaves on calabi-yau manifolds

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

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

Pro<strong>of</strong>. All the conditions <strong>of</strong> Theorem 2.2.8 are satisfied, for A = Mod(X, α), A ′ = Coh(X, α), B = Mod(X h , α h ), B ′ = Coh(X h , α h ) and F . (Use Theorem 1.3.9 and Proposition 2.2.9.) For the last statement, use again dévissage; the basis <strong>of</strong> the induction is provided by the fact that h ∗ Ext i Coh(X,α)(F , G ) ∼ = Ext i Coh(Xh ,αh ) (F (F ), F (G )) for any F ∈ Coh(X, α), G ∈ Coh(X, α ′ ). 2.3 Relations Among Derived Functors The pro<strong>of</strong>s <strong>of</strong> the following statements are entirely similar to those <strong>of</strong> the corresponding ones in [23, II.5], so they will be omitted. Note that the results are given in their full generality, but we will only need them in a much more restricted situation: all the spaces involved are smooth and compact, all morphisms are proper and smooth, all twistings are in the Brauer group and all complexes are bounded. These results hold then with no extra restrictions. Proposition 2.3.1. Let f : X → Y and g : Y → Z be two proper morphisms <strong>of</strong> schemes or analytic spaces, and let α ∈ ˇ H2 (Z, O∗ Z ). Then there is a natural isomorphism R(g∗ ◦ f∗) ∼ −→ Rg∗ ◦ Rf∗ <strong>of</strong> functors from Dcoh(X, f ∗ g ∗ α) to Dcoh(Z, α). Pro<strong>of</strong>. To prove this, all we need to do is show that if I is an injective f ∗ g ∗ αsheaf, then f∗I is a g∗-acyclic g ∗ α-sheaf. But this statement is local on Z, and therefore it follows from the corresponding one for <strong>sheaves</strong> by Lemma 1.2.6. Proposition 2.3.2. Let X be a scheme or analytic space, and let α ∈ ˇ H 2 (X, O ∗ X ). Then there is a natural isomorphism R Hom · (F · , G · ) ∼ −→ RΓ(X, RHom · (F · , G · )) <strong>of</strong> bi-functors from D − coh (X, α)◦ × D + coh (X, α) to D(Ab). Pro<strong>of</strong>. The pro<strong>of</strong> follows as in [23, I.5.3], using the easy fact that if I is an injective α-sheaf, Hom(F , I ) is flasque for any α-sheaf F (easy). Proposition 2.3.3. Let f : X → Y and g : Y → Z be morphisms <strong>of</strong> schemes or analytic spaces, and let α ∈ ˇ H2 (Z, O∗ Z ). Then there is a natural isomorphism L(f ∗ ◦ g ∗ ) ∼ −→ Lf ∗ ◦ Lg ∗ <strong>of</strong> functors from D − coh (Z, α) to D− coh (X, f ∗g∗α). 34
Proposition 2.3.4. Let f : X → Y be a proper morphism <strong>of</strong> schemes or analytic spaces, and let α, α ′ ∈ ˇ H2 (Y, O∗ Y ). Then there is a natural functorial homomorphism Rf∗RHom · X(F · , G · ) −→ RHom · Y (Rf∗F · , Rf∗G · ) for F · ∈ D − coh (X, f ∗ α), G · ∈ D + coh (X, f ∗ α ′ ). Proposition 2.3.5 (Projection Formula). Let f : X → Y be a proper morphism <strong>of</strong> schemes or analytic spaces, and let α, α ′ ∈ ˇ H2 (Y, O∗ Y ). Then there is a natural functorial isomorphism Rf∗(F · ) L ⊗Y G · ∼ L · −→ Rf∗(F ⊗X Lf ∗ G · ) for F · ∈ D − coh (X, f ∗ α) and G · ∈ D − coh (Y, α′ ). Proposition 2.3.6. Let f : X → Y be a flat morphism <strong>of</strong> schemes or analytic spaces, and let α, α ′ ∈ ˇ H2 (Y, O∗ Y ). Then there is a natural functorial isomorphism f ∗ RHom · Y (F · , G · ) ∼ −→ RHom · X(f ∗ F · , f ∗ G · ) for F · ∈ D − coh (Y, α), G· ∈ D + coh (Y, α′ ). (We write f ∗ instead <strong>of</strong> Lf ∗ because it is an exact functor.) Proposition 2.3.7. Let f : X → Y be a morphism <strong>of</strong> schemes or analytic spaces, and let α, α ′ ∈ ˇ H2 (Y, O∗ Y ). Then there is a natural functorial isomorphism for F · ∈ D − coh (Y, α), G· ∈ D − coh (Y, α′ ). Lf ∗ (F · ) L ⊗X Lf ∗ (G · ) ∼ −→ Lf ∗ L · (F ⊗Y G · ) Proposition 2.3.8. Let f : X → Y be a proper morphism <strong>of</strong> schemes or analytic spaces, and let α, α ′ ∈ ˇ H2 (Y, O∗ Y ). Then there is a natural functorial homomorphism F · → Rf∗Lf ∗ F · for F · ∈ D − coh (Y, α) which gives rise by proposition 2.3.4 to a natural functorial isomorphism Rf∗RHom · X(Lf ∗ F · , G · ) ∼ −→ RHom · Y (F · , Rf∗G · ) for F · ∈ D − coh (Y, α), G· ∈ D + coh (X, f ∗ α ′ ). Corollary 2.3.9 (Adjoint property <strong>of</strong> f∗ and f ∗ ). If f : X → Y is a proper morphism <strong>of</strong> schemes or analytic spaces, and if α ∈ ˇ H2 (Y, O∗ X ) then we have HomDcoh(X,f ∗ α)(Lf ∗ F · , G · ) ∼ −→ HomDcoh(Y,α)(F · , Rf∗G · ) for F · ∈ D − coh (Y, α), G· ∈ D + coh (X, f ∗ α ′ ). In other words Lf ∗ and Rf∗ are adjoint functors from D − coh respectively. (Y, α) to D− coh (X, f ∗ α) and D + coh (X, f ∗ α) to D + coh 35 (Y, α),
Page 1 and 2: DERIVED CATEGORIES OF TWISTED SHEAV
Page 3 and 4: DERIVED CATEGORIES OF TWISTED SHEAV
Page 5 and 6: Tuturor celor din care am fost făc
Page 7 and 8: Table of Contents
Page 9 and 10: List of Figures 6.
Page 11 and 12: Introduction and Overview Twisted <
Page 13 and 14: twisted by α ∈
Page 15 and 16: fibers introduces interesting featu
Page 17 and 18: Chapter 1 Twisted Sheaves In this c
Page 19 and 20: to be H2 an(X, O∗ X ). If we do n
Page 21 and 22: is bijective. Then A is called an A
Page 23 and 24: such that α = δa and β = δb. Si
Page 25 and 26: Lemma 1.2.3. Let α ∈ Č2 (X, O
Page 27 and 28: Proof. Let U ′
Page 29 and 30: Proof. Since E and
Page 31 and 32: Proof. Follows fro
Page 33 and 34: Definition 1.3.12. Let A be a ring.
Page 35 and 36: Definition 1.3.18. Two R-algebras A
Page 37 and 38: property “free on stalks” for t
Page 39 and 40: 2.2 The Derived Category and Derive
Page 41 and 42: to reduce to the case when F · als
Page 43: So assume n > 0, and as before cons
Page 47 and 48: 2.4 Duality for Proper Smooth Morph
Page 49 and 50: 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 and 102:
contract 45 curves 2H+D-E contract
Page 103 and 104:
Remark 6.3.2. This theorem basicall
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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
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113 One space of i
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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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