derived categories of twisted sheaves on calabi-yau manifolds

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

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

<strong>of</strong> J is to view it as a relative moduli space (over S) <strong>of</strong> semistable <strong>sheaves</strong> on the fibers <strong>of</strong> X, <strong>of</strong> rank one and degree zero. (This may seem like a contorted way to say “degree zero line bundles”, but it will pay <strong>of</strong>f when we’ll consider non-smooth fibrations.) Since J is a moduli space <strong>of</strong> stable <strong>sheaves</strong> on X, we obtain a twisting α ∈ Br(J) and a <strong>twisted</strong> universal sheaf U on X ×S J (the product is the fibered product, since everything we do is relative to the base S). Recall that in the case <strong>of</strong> smooth elliptic fibrations there is a well-known identification <strong>of</strong> Br(J)/ Br(S) with the Tate-Shafarevich group XS(J). The main property <strong>of</strong> XS(J) is that it classifies elliptic fibrations (possibly without a section) whose relative Jacobian is J. Having started with an elliptic fibration X → S whose Jacobian is J, we have thus defined two different elements <strong>of</strong> XS(J): the image <strong>of</strong> α, that measures the obstruction to the existence <strong>of</strong> a universal sheaf on X ×S J, and αX/S, which represents X → S in XS(J) = Br(J)/ Br(S). A first result in the study <strong>of</strong> these objects is Theorem 4.4.1: α mod Br(S) = αX/S. In other words, we can recover Ogg-Shafarevich theory from the theory <strong>of</strong> the obstructions to the existence <strong>of</strong> a universal sheaf for a moduli problem. Next we move on to K3 surfaces, and we study moduli spaces <strong>of</strong> <strong>sheaves</strong> on them. We are mainly interested in the case when the moduli space is twodimensional, in which case we know that it is again a K3 surface. Building upon the work <strong>of</strong> Mukai, we are able to identify the obstruction to the existence <strong>of</strong> a universal sheaf, only in terms <strong>of</strong> the cohomological Fourier-Mukai transform. This provides an interesting interpretation <strong>of</strong> the <strong>twisted</strong> <strong>derived</strong> <strong>categories</strong> <strong>of</strong> a K3 X: if the usual <strong>derived</strong> category corresponds (by a result <strong>of</strong> Orlov, [36]) to the Hodge structure on the transcendental lattice TX <strong>of</strong> X, then the <strong>derived</strong> category <strong>of</strong> α-<strong>twisted</strong> <strong>sheaves</strong> on X corresponds to the induced Hodge structure on the sublattice Ker(α) <strong>of</strong> TX. (This makes sense, because for a K3 surface we have an identification Br(X) ∼ = T ∨ X ⊗ Q/Z, where T ∨ X denotes the dual lattice to TX.) In particular, this allows us to conclude that in a number <strong>of</strong> interesting cases we have D b coh(X, α) ∼ = D b coh(X, α k ) for a K3 X, α ∈ Br(X), and k ∈ Z coprime to the order <strong>of</strong> α. This result is quite striking: although a similar statement can be made over any scheme, it could not hold over the spectrum <strong>of</strong> any local ring! This result therefore illustrates the global nature <strong>of</strong> <strong>derived</strong> <strong>categories</strong>. The next chapter deals with elliptic Calabi-Yau threefolds. We only restrict our attention to the simplest examples, where the singularities <strong>of</strong> the fibers are easiest to understand (see Chapter 6 for details). Birationally, these are the same as the smooth elliptic threefolds studied previously, but the existence <strong>of</strong> singular 4
fibers introduces interesting features. Although the starting space X is smooth, the relative Jacobian J has singularities, and these cannot be removed in the algebraic category without losing the Calabi-Yau property KJ = 0. The solution to this is to work in the analytic category; we prove that for any analytic small resolution ¯J → J <strong>of</strong> the singularities <strong>of</strong> J we can find a <strong>twisted</strong> pseudo-universal sheaf U on X ×S ¯ J. (S is the base for the fibrations under consideration.) The reason we call this <strong>twisted</strong> sheaf “pseudo-universal” is the fact that over the stable points <strong>of</strong> ¯ J (which lie over the smooth points <strong>of</strong> J) U is equal to the <strong>twisted</strong> universal sheaf whose existence was asserted before. However, over points <strong>of</strong> ¯ J that lie over a singular – and semistable – point P <strong>of</strong> J, the sheaf U parametrizes some <strong>of</strong> the semistable <strong>sheaves</strong> in the S-equivalence class whose image in J is P . Using this <strong>twisted</strong> pseudo-universal sheaf U , we can define a Fourier-Mukai equivalence D b coh(X) ∼ = D b coh( ¯ J, α −1 ), where α ∈ Br ′ ( ¯ J) ⊆ Br ′ (J) unique extension to ¯ J <strong>of</strong> the obstruction to the existence <strong>of</strong> a universal sheaf on X ×S J s . This allows us to again construct a number <strong>of</strong> examples in which we have D b coh( ¯ J, α) ∼ = D b coh( ¯ J, α k ) for k coprime to the order <strong>of</strong> α. This seems to be a rather general phenomenon for <strong>twisted</strong> <strong>sheaves</strong> on schemes or complex manifolds! One section <strong>of</strong> the chapter on elliptic Calabi-Yau threefolds is dedicated to applying the previously obtained results on cohomological Fourier-Mukai transforms. We are able to produce two elliptic Calabi-Yau threefolds which are not birational, but whose Z-Hodge structures and cup-product structures are identical. This provides a counterexample to the generalization <strong>of</strong> the Torelli theorem from K3 surfaces to Calabi-Yau threefolds. (For K3’s, the corresponding statement is that if the Hodge structures on the cohomologies <strong>of</strong> two K3’s are the same, respecting the cup product, then the K3’s are isomorphic. It was known before that there can be non-isomorphic Calabi-Yau’s with isomorphic Z-Hodge structures, but it was believed that they should at least be birational to each other. We are unable to prove that the two Calabi-Yau’s we produce are not birational to each other, but we give strong supporting evidence that they are not.) This construction has another interesting side to it: the apparition <strong>of</strong> the singularities in J seems to be caused by the same phenomenon that causes singularities to appear in an example <strong>of</strong> Vafa-Witten ([43]) and Aspinwall-Morrison ([1]). We give a brief suggestion as to why these singularities are unavoidable in Section 6.8. The dissertation closes with a chapter in which we state a number <strong>of</strong> open questions, that would deserve further investigation. Hopefully, these questions will attract the interest <strong>of</strong> someone with mightier mathematical powers than mine. . . 5
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: twisted by α ∈
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 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</
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Jij → S. This means that if Fij w
Page 67 and 68:
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)
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Theorem 5.1.10. Under the hypothese
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groups of differen
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Let c be the image of</stro
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Proof. Trivial cha
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and therefore rk(V ) = (v(V ), (0,
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for some k ≫ 0. By the same argum
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and therefore Now consider the map
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In order to identify the kernel, no
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for all F , G stable sheave
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Chapter 6 Elliptic Calabi-Yau Three
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Definition 6.1.2. A projective morp
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Example 6.2.1. We’ve already seen
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contract 45 curves 2H+D-E contract
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Remark 6.3.2. This theorem basicall
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But this is impossible, because the
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We have H 0 (C, IQ) = 0 and χ(IQ)
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Having fixed a smooth point P ∈ C
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101 Proof. Since X
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103 Therefore, α can be represente
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Corollary 6.5.6. Under the hypothes
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107 Remark 6.6.4. We could also pro
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109 where the intersection form on
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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
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Bibliography [1] Aspinwall, P., Mor
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119 [28] D. Bayer and M. Stillman,
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derived categories of twisted sheaves on calabi-yau manifolds

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