derived categories of twisted sheaves on calabi-yau manifolds

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

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

Proposition 2.2.5 (The Hypercohomology Spectral Sequence). Let A, B be abelian <strong>categories</strong>, and let F : A → B be an additive, left exact functor. Assume that A has enough injectives, so that the <strong>derived</strong> functor exists. In particular the functors RF : D + (A) → D + (B) R i F : A → B are defined. Let X · be a complex in D + (A). Then there exists a spectral sequence E i,j k , such that E i,j 2 = R i F (H j (X · )), and such that E i,j k ⇒ Hi+j (RF (X · )). If the functor F has finite cohomological dimension on A (i.e. there exists a fixed n such that we have R i F (A) = 0 for all A ∈ A and all i > n) then the above statement holds with D + replaced by D. Pro<strong>of</strong>. See [30, Appendix C]. Theorem 2.2.6. Under the assumptions <strong>of</strong> Theorem 2.2.4, assume furthermore that X and Y are smooth <strong>of</strong> finite dimension, and that the morphism f is proper. Then the following <strong>derived</strong> functors are defined: RHom · : D b coh(X, α) ◦ × D b coh(X, α ′ ) → D b coh(X, α −1 α ′ ), L ⊗ : D b coh(X, α) × D b coh(X, α ′ ) → D b coh(X, αα ′ ), Lf ∗ : D b coh(X, α) → D b coh(Y, f ∗ α), Rf∗ : D b coh(Y, f ∗ α) → D b coh(X, α). If X is any scheme, or is a compact complex analytic space, then is also defined. R Hom · : D b coh(X, α) ◦ × D b coh(X, α) → D b coh(Ab) Pro<strong>of</strong>. The only thing we need to do is prove that all the functors defined previously take complexes with bounded cohomology to complexes with bounded cohomology. For Rf∗, Lf ∗ and L ⊗ the result follows immediately from Proposition 2.2.5. Indeed, using Proposition 2.1.8 one finds that the E i,j 2 terms <strong>of</strong> the hypercohomology spectral sequence are contained in a bounded rectangle <strong>of</strong> the (i, j)-plane, and therefore, after converging, the cohomology <strong>of</strong> the total complex is bounded. Now consider the case <strong>of</strong> RHom · (F · , G · ) for two bounded complexes F · ∈ D b coh (X, α), G· ∈ D b coh (X, α′ ). Using again Proposition 2.2.5 we reduce to the case when G · consists <strong>of</strong> a single sheaf. We’ll use a technique known as dévissage 30
to reduce to the case when F · also consists <strong>of</strong> a single sheaf, which is just Proposition 2.1.6. The dévissage technique is just induction on the number n(F · ) defined as n(F · ) = max{j − i | H j (F · ) = 0, H i (F · ) = 0}. The basis <strong>of</strong> the induction is the case n(F · ) = 0, when F · is quasi-isomorphic to a single sheaf, so that Proposition 2.1.6 gives the boundedness <strong>of</strong> RHom · (F · , G · ). Now assume proven that RHom · (F · , G · ) is bounded for n(F · ) ≤ n0 (for some n0 ≥ 0). Phrased differently, this means that we have Ext i (F · , G · ) = 0 for any F · with n(F · ) ≤ n0 and for all |i| ≫ 0. Assume that F · has n(F · ) = n0 + 1. Translating F · , we can assume that H i (F · ) = 0 for i < 0, and H 0 (F · ) = 0. Using Remark 2.2.3 we can assume that in fact F i = 0 for i < 0. Let F ′· be the complex that consists <strong>of</strong> only one sheaf equal to H 0 (F · ), in degree 0, and consider the natural map <strong>of</strong> complexes F ′· → F · which is an isomorphism on H 0 , and zero on the other cohomologies. Fit this morphism into a triangle F ′· ✲ F · ■❅ ❅❅ ✠ F ′′· , and write down the long exact Ext( · , G · )-sequence for this triangle, obtained as in [23, I.6.1]. Note that we do not need locally free resolutions for that, the pro<strong>of</strong> being entirely similar to that in [22, III.6.4]. Since we have n(F ′′· ) < n(F · ) (by the long exact cohomology sequence for the above triangle), we can use the induction hypothesis that Ext i (F ′′· , G · ) = 0 for |i| ≫ 0 as well as the fact that Ext i (F ′· , G · ) = 0 for |i| ≫ 0, to conclude that Ext i (F · , G · ) = 0 for |i| ≫ 0, which is what we wanted. The case <strong>of</strong> R Hom · will follow from the fact that we have R Hom · (F · , G · ) = RΓ(RHom · (F · , G · )), (Proposition 2.3.2) and the fact that on a scheme or on a compact analytic space RΓ takes bounded complexes to bounded complexes. Remark 2.2.7. Note that we have to go through all this tortuous process just because we do not know if locally free resolutions exist. If they did, a criterion entirely similar to that in Proposition 2.1.8, combined with a hypercohomology spectral sequence, would allow us to solve the problem as we did for L ⊗. As another application <strong>of</strong> the dévissage technique, we prove the following theorem, which will be used to prove an analogue <strong>of</strong> GAGA (2.2.10). 31
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: 2.2 The Derived Category and Derive
Page 43 and 44: So assume n > 0, and as before cons
Page 45 and 46: Proposition 2.3.4. Let f : X → Y
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
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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