Twisted Hodge diamonds give rise to non-Fourier-Mukai functors
Felix Küng
备注：
45 Pages, minor typos corrected, changed internal referencing to make use of assumptions clearer, improved clarity of arguments using generators. Accepted to be published in "Journal of Noncommutative Geometry". Comments are welcome, please email to felix.kung@ulb.be
我们应用扭曲霍奇菱形的计算来构造无穷多个具有良好目标空间和源空间的非傅里叶-向井函子。为此，我们首先研究特征态射，以便对倾斜丛进行控制。接着，我们应用嵌入射影空间中的超曲面的扭曲霍奇菱形来计算这些空间的霍赫希尔德维数。这使我们能够计算嵌入到射影空间中的映射在霍赫希尔德上同调中的核。最后，我们利用上述计算来应用A. Rizzardo、M. Van den Bergh和A. Neeman关于非傅里叶-向井函子的构造，并验证所构造的函子对于奇数维二次超曲面确实不可能是傅里叶-向井函子。通过这种方法，我们证明了存在大量可用于此类构造的霍赫希尔德上同调类。此外，我们的结果允许应用基于计算机的计算来为任意高维中的任意次数超曲面构造候选函子。验证这些函子不是傅里叶-向井函子仍然需要倾斜丛的存在性。特别地，我们证明了对于每一个奇数维光滑二次超曲面，至少存在一个非傅里叶-向井函子。
英文摘要：
We apply computations of twisted Hodge diamonds to construct an infinite number of non-Fourier-Mukai functors with well behaved target and source spaces. To accomplish this we first study the characteristic morphism in order to control it for tilting bundles. Then we continue by applying twisted Hodge diamonds of hypersurfaces embedded in projective space to compute the Hochschild dimension of these spaces. This allows us to compute the kernel of the embedding into the projective space in Hochschild cohomology. Finally we use the above computations to apply the construction by A. Rizzardo, M. Van den Bergh and A. Neeman of non-Fourier-Mukai functors and verify that the constructed functors indeed cannot be Fourier-Mukai for odd dimensional quadrics. Using this approach we prove that there are a large number of Hochschild cohomology classes that can be used for this type construction. Furthermore, our results allow the application of computer-based calculations to construct candidate functors for arbitrary degree hypersurfaces in arbitrary high dimensions. Verifying that these are not Fourier-Mukai still requires the existence of a tilting bundle. In particular we prove that there is at least one non-Fourier-Mukai functor for every odd dimensional smooth quadric.
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We describe the birational correspondences, induced by the Fourier-Mukai functor, between moduli spaces of semistable sheaves on elliptic surfaces with sections, using the notion of $P$-stability in the derived category. We give explicit conditions to determine whether these correspondences are isomorphisms. This is indeed not true in general and we describe the cases where the birational maps are Mukai flops. Moreover, this construction provides examples of new compactifications of the moduli spaces of vector bundles via sheaves with torsion and via complexes. We finally get for any fixed dim…
Beauville and Donagi proved in 1985 that the primitive middle cohomology of a smooth complex cubic fourfold and the primitive second cohomology of its variety of lines, a smooth hyperkähler fourfold, are isomorphic as polarized integral Hodge structures. We prove analogous statements for smooth complex Gushel-Mukai varieties of dimension 4 (resp. 6), i.e., smooth dimensionally transverse intersections of the cone over the Grassmannian Gr(2,5), a quadric, and two hyperplanes (resp. of the cone over Gr(2,5) and a quadric). The associated hyperkähler fourfold is in both cases a smooth double cove…
We give an introductory review of Fourier-Mukai transforms and their application to various aspects of moduli problems, string theory and mirror symmetry. We develop the necessary mathematical background for Fourier-Mukai transforms such as aspects of derived categories and integral functors as well as their relative version which becomes important for making precise the notion of fiberwise T-duality on elliptic Calabi-Yau threefolds. We discuss various applications of the Fourier-Mukai transform to D-branes on Calabi-Yau manifolds as well as homological mirror symmetry and the construction of…
Given a scheme in characteristic p together with a lifting modulo p^2, we construct a functor from a category of suitably nilpotent modules with connection to the category of Higgs modules. We use this functor to generalize the decomposition theorem of Deligne-Illusie to the case of de Rham cohomology with coefficients.
We study the derived categories of twisted supersingular K3 surfaces. We prove a derived crystalline Torelli theorem for twisted supersingular K3 surfaces, characterizing Fourier-Mukai equivalences in terms of isomorphisms between their associated K3 crystals. This is a positive characteristic analog of the Hodge-theoretic derived Torelli theorem of Orlov, and its extension to twisted K3 surfaces by Huybrechts and Stellari. We give applications to various questions concerning Fourier-Mukai partners, extending results of Căldăraru and Huybrechts and Stellari. We also give an exact formula for t…
Let X be an abelian scheme over a scheme B. The Fourier--Mukai transform gives an equivalence between the derived category of X and the derived category of the dual abelian scheme. We partially extend this to certain schemes X over B (which we call degenerate abelian schemes) whose generic fiber is an abelian variety, while special fibers are singular. Our main result provides a fully faithful functor from a twist of the derived category of Pic$^τ$(X/B) to the derived category of X. Here Pic$^τ$(X/B) is the algebraic space classifying fiberwise numerically trivial line bundles. Next, we show t…
In this paper, we consider moduli spaces of stable sheaves on abelian surfaces. Our main assumption is the primitivity of the associated Mukai vector. We construct many isomorphisms of muduli spaces induced by Fourier-Mukai functor. As an application, we show that deformation type of these spaces are determined by their dimension. We next show that the fiber of the albanese map is irreducible symplectic manifolds In particular, we describe the period by using Mukai lattice. We also discuss deformation type of moduli spaces of stable sheaves on K3 surfaces.
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For $X$ a smooth quasi-projective variety and $X^{[n]}$ its associated Hilbert scheme of $n$ points, we study two canonical Fourier--Mukai transforms $D(X)\to D(X^{[n]})$, the one along the structure sheaf and the one along the ideal sheaf of the universal family. For $\dim X\ge 2$, we prove that both functors admit a left-inverse. This means in particular that both functors are faithful and injective on isomorphism classes of objects. Using another method, we also show in the case of an elliptic curve that the Fourier--Mukai transform along the structure sheaf of the universal family is faith…
