Fourier-Mukai Partners Research Articles

Birational geometry of Beauville–Mukai systems III: Asymptotic behavior

AbstractSuppose that a Hilbert scheme of points on a K3 surface of Picard rank one admits a rational Lagrangian fibration. We show that if the degree of the surface is sufficiently large compared to the number of points, then the Hilbert scheme is the unique hyperkähler manifold in its birational class. In particular, the Hilbert scheme is a Lagrangian fibration itself, which we realize as coming from a (twisted) Beauville–Mukai system on a Fourier–Mukai partner of . We also show that when the degree of the surface is small our method can be used to find all birational models of the Hilbert scheme.

Autoequivalences of blow‐ups of minimal surfaces

AbstractLet be the blow‐up of in a finite set of very general points. We deduce from the work of Uehara [Trans. Amer. Math. Soc. 371 (2019), no. 5, 3529–3547] that has only standard autoequivalences, no non‐trivial Fourier–Mukai partners, and admits no spherical objects. If is the blow‐up of in 9 very general points, we provide an alternate and direct proof of the corresponding statement. Further, we show that the same result holds if is a blow‐up of finitely many points in a minimal surface of non‐negative Kodaira dimension which contains no ‐curves. Independently, we characterize spherical objects on blow‐ups of minimal surfaces of positive Kodaira dimension.

Derived Equivalence for Elliptic K3 Surfaces and Jacobians

Abstract We present a detailed study of elliptic fibrations on Fourier-Mukai partners of K3 surfaces, which we call derived elliptic structures. We fully classify derived elliptic structures in terms of Hodge-theoretic data, similar to the Derived Torelli Theorem that describes Fourier-Mukai partners. In Picard rank two, derived elliptic structures are fully determined by the Lagrangian subgroups of the discriminant group. As a consequence, we prove that for a large class of Picard rank 2 elliptic K3 surfaces all Fourier-Mukai partners are Jacobians, and we partially extend this result to non-closed fields. We also show that there exist elliptic K3 surfaces with Fourier-Mukai partners, which are not Jacobians of the original K3 surface. This gives a negative answer to a question raised by Hassett and Tschinkel.

A note on Fourier–Mukai partners of abelian varieties over positive characteristic fields

A note on Fourier–Mukai partners of abelian varieties over positive characteristic fields

Fourier–Mukai partners for very general special cubic fourfolds

We exhibit explicit examples of very general special cubic fourfolds with discriminant $d$ admitting an associated (twisted) K3 surface, which have non-isomorphic Fourier-Mukai partners. In particular, in the untwisted setting, we show that the number of Fourier-Mukai partners for a very general special cubic fourfold with discriminant $d$ and having an associated K3 surface, is equal to the number $m$ of Fourier-Mukai partners of its associated K3 surface, if $d \equiv 2 (\text{mod}\,6)$; else, if $d \equiv 0 (\text{mod}\,6)$, the cubic fourfold has $\lceil m/2 \rceil$ Fourier-Mukai partners.

Fourier–Mukai partners of Enriques and bielliptic surfaces in positive characteristic

We prove that a twisted Enriques (respectively, untwisted bielliptic) surface over an algebraically closed field of positive characteristic at least 3 (respectively, at least 5) has no non-trivial Fourier-Mukai partners.

Derived equivalences of twisted supersingular K3 surfaces

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 the twisted K3 crystals introduced in [2]. This is a positive characteristic analog of the Hodge-theoretic derived Torelli theorem of Orlov [23] and its extension to twisted K3 surfaces by Huybrechts and Stellari [9,10]. We give applications to various questions concerning Fourier-Mukai partners, extending results of Căldăraru [5] and Huybrechts and Stellari [9]. We also give an exact formula for the number of twisted Fourier-Mukai partners of a twisted supersingular K3 surface.

Twisted Fourier–Mukai partners of Enriques surfaces

Bridgeland and Maciocia showed that a complex Enriques surface X has no Fourier-Mukai partners apart from itself: that is, if D^b(X) = D^b(Y) then X = Y. We extend this to twisted Fourier-Mukai partners: if alpha is the non-trivial element of Br(X) = Z/2 and D^b(X,alpha) = D^b(Y,beta), then X = Y and beta is non-trivial. Our main tools are twisted topological K-theory and twisted Mukai lattices.

Derived equivalences of canonical covers of hyperelliptic and Enriques surfaces in positive characteristic

We prove that any Fourier–Mukai partner of an abelian surface over an algebraically closed field of positive characteristic is isomorphic to a moduli space of Gieseker-stable sheaves. We apply this fact to show that the set of Fourier–Mukai partners of a canonical cover of a hyperelliptic or Enriques surface over an algebraically closed field of characteristic greater than three is trivial. These results extend earlier results of Bridgeland–Maciocia and Sosna to positive characteristic.

On Derived Equivalences of K3 Surfaces in Positive Characteristic

For an ordinary K3 surface over an algebraically closed field of positive characteristic we show that every automorphism lifts to characteristic zero. Moreover, we show that the Fourier-Mukai partners of an ordinary K3 surface are in one-to-one correspondence with the Fourier-Mukai partners of the geometric generic fiber of its canonical lift. We also prove that the explicit counting formula for Fourier-Mukai partners of the K3 surfaces with Picard rank two and with discriminant equal to minus of a prime number, in terms of the class number of the prime, holds over a field of positive characteristic as well. We show that the image of the derived autoequivalence group of a K3 surface of finite height in the group of isometries of its crystalline cohomology has index at least two. Moreover, we provide a conditional upper bound on the kernel of this natural cohomological descent map. Further, we give an extended remark in the appendix on the possibility of an F-crystal structure on the crystalline cohomology of a K3 surface over an algebraically closed field of positive characteristic and show that the naive F-crystal structure fails in being compatible with inner product.

Minuscule Schubert Varieties and Mirror Symmetry

We consider smooth complete intersection Calabi-Yau 3-folds in minuscule Schubert varieties, and study their mirror symmetry by degenerating the ambient Schubert varieties to Hibi toric varieties. We list all possible Calabi-Yau 3-folds of this type up to deformation equivalences, and find a new example of smooth Calabi-Yau 3-folds of Picard number one; a complete intersection in a locally factorial Schubert variety ${\boldsymbol{\Sigma}}$ of the Cayley plane ${\mathbb{OP}}^2$. We calculate topological invariants and BPS numbers of this Calabi-Yau 3-fold and conjecture that it has a non-trivial Fourier-Mukai partner.

Derived equivalences and Kodaira fibers

We give necessary conditions for two (including non-reduced and multiple) Kodaira curves to be derived equivalent. We classify Fourier-Mukai partners of any reduced Kodaira curve. We prove that the derived category of singularities of any non-reduced and non-multiple Kodaira curve is idempotent complete.

Fourier–Mukai partners of elliptic ruled surfaces

We study Fourier–Mukai partners of elliptic ruled surfaces. We also describe the autoequivalence group of the derived categories of ruled surfaces with an elliptic fibration, by using another work of the author.

Weighted noncommutative regular projective curves

Let \mathcal H be a noncommutative regular projective curve over a perfect field k . We study global and local properties of the Auslander–Reiten translation \tau and give an explicit description of the complete local rings, with the involvement of \tau . We introduce the \tau -multiplicity e_{\tau}(x) , the order of \tau as a functor restricted to the tube concentrated in x . We obtain a local-global principle for the (global) skewness s(\mathcal H) , defined as the square root of the dimension of the function (skew-) field over its centre. In the case of genus zero we show how the ghost group, that is, the group of automorphisms of \mathcal H which fix all objects, is determined by the points x with e_{\tau}(x) > 1 . Based on work of Witt we describe the noncommutative regular (smooth) projective curves over the real numbers; those with s(\mathcal H)=2 we call Witt curves. In particular, we study noncommutative elliptic curves, and present an elliptic Witt curve which is a noncommutative Fourier–Mukai partner of the Klein bottle. If \mathcal H is weighted, our main result will be formulae for the orbifold Euler characteristic, involving the weights and the \tau -multiplicities. As an application we will classify the noncommutative 2-orbifolds of nonnegative Euler characteristic, that is, the real elliptic, domestic and tubular curves. Throughout, many explicit examples are discussed.

Fourier-Mukai partners of K3 surfaces in positive characteristic

We study Fourier-Mukai equivalence of K3 surfaces in positive characteristic and show that the classical results over the complex numbers all generalize. The key result is a positive-characteristic version of the Torelli theorem that uses the derived category in place of the Hodge structure on singular cohomology; this is proven by algebraizing formal lifts of Fourier-Mukai kernels to characteristic zero. As a consequence, any Shioda-supersingular K3 surface is uniquely determined up to isomorphism by its derived category of coherent sheaves. We also study different realizations of Mukai's Hodge structure in algebraic cohomology theories (etale, crystalline, de Rham) and use these to prove: 1) the zeta function of a K3 surface is a derived invariant (discovered independently by Huybrechts); 2) the variational crystalline Hodge conjecture holds for correspondences arising from Fourier-Mukai kernels on products of two K3 surfaces.

Fourier–Mukai partners of singular genus one curves

The objective of the paper is to prove that, as it happens for smooth elliptic curves, any Fourier-Mukai partner of a projective reduced Gorenstein curve of genus one and trivial dualising sheaf, is isomorphic to itself.

Fourier-Mukai Partners of Canonical Covers of Bielliptic and Enriques Surfaces

We prove that the canonical cover of an Enriques surface does not admit non-trivial Fourier-Mukai partners. We also show that the canonical cover of a bielliptic surface has at most one non-isomorphic Fourier-Mukai partner. The first result is then applied to birational Hilbert schemes of points and the second to birational generalised Kummer varieties

Relative Fourier–Mukai transforms for Weierstraß fibrations, abelian schemes and Fano fibrations

AbstractWe study the group of relative Fourier–Mukai transforms for Weierstraß fibrations, abelian schemes and Fano or anti-Fano fibrations. For Weierstraß and Fano or anti-Fano fibrations we describe this group completely. For abelian schemes over an arbitrary base we prove that if two of them are relative Fourier–Mukai partners then there is an isometric isomorphism between the fibre products of each of them and its dual abelian scheme. If the base is normal and the slope map is surjective we show that these two conditions are equivalent. Moreover in this situation we completely determine the group of relative Fourier–Mukai transforms and we prove that the number of relative Fourier–Mukai partners of a given abelian scheme over a normal base is finite.

Reconstruction and finiteness results for Fourier–Mukai partners

We develop some methods for studying the Fourier–Mukai partners of an algebraic variety. As applications we prove that abelian varieties have finitely many Fourier–Mukai partners and that they are uniquely determined by their derived category of coherent D-modules. We also generalize a famous theorem due to Bondal and Orlov.

Cusps of the Kähler moduli space and stability conditions on K3 surfaces

Ma (Int J Math 20(6):727–750, 2009) established a bijection between Fourier–Mukai partners of a K3 surface and cusps of the Kähler moduli space. The Kähler moduli space can be described as a quotient of Bridgeland’s stability manifold. We study the relation between stability conditions σ near to a cusp and the associated Fourier–Mukai partner Y in the following ways. (1) We compare the heart of σ to the heart of coherent sheaves on Y. (2) We construct Y as moduli space of σ-stable objects. An appendix is devoted to the group of auto-equivalences of \({\mathcal{D}^b(X)}\) which respect the component \({Stab^{\dagger}(X)}\) of the stability manifold.