Gushel-Mukai varieties are smooth complex dimensionally transverse intersections of a cone over the Grassmannian $\mathsf{Gr}(2,5)$ with a linear space and a quadratic hypersurface. The aim of this survey is to discuss the geometry, moduli, Hodge structures, and categorical aspects of these varieties. It is based on joint work with Alexander Kuznetsov and earlier work of Logachev, Iliev, Manivel, O'Grady, and others.
We construct an explicit complex smooth Fano threefold with Picard number 1, index 1, and degree 10 (also known as a Gushel-Mukai threefold) and prove that it is not rational by showing that its intermediate Jacobian has a faithful $\mathrm{PSL}(2,\mathbf{F}_{11}) $-action. Along the way, we construct Gushel-Mukai varieties of various dimensions with rather large (finite) automorphism groups. The starting point of all these constructions is an Eisenbud-Popescu-Walter sextic with a faithful $\mathrm{PSL}(2,\mathbf{F}_{11}) $-action discovered by the second author in 2013.
We provide a stable rationality construction for some smooth complex Gushel-Mukai varieties of dimension 6. As a consequence, we compute the integral singular cohomology of any smooth Gushel-Mukai sixfold and in particular, show that it is torsion-free.
We construct new examples of rational Gushel-Mukai fourfolds, giving more evidence for the analog of the Kuznetsov Conjecture for cubic fourfolds: a Gushel--Mukai fourfold is rational if and only if it admits an associated K3 surface.
In the first part of this paper we will prove the Voevodsky's nilpotence conjecture for smooth cubic fourfolds and ordinary generic Gushel-Mukai fourfolds. Then, making use of noncommutative motives, we will prove the Voevodsky's nilpotence conjecture for generic Gushel-Mukai fourfolds containing a $τ$-plane $\G(2,3)$ and for ordinary Gushel-Mukai fourfolds containing a quintic del Pezzo surface.
We prove that the moduli stacks of marked and labelled Hodge-special Gushel-Mukai fourfolds are isomorphic. As an application, we construct rational maps from the stack of Hodge-special Gushel-Mukai fourfolds of discriminant $d$ to the moduli space of (twisted) degree-$d$ polarized K3 surfaces. We use these results to prove a counting formula for the number of 4-dimensional fibers of Fourier-Mukai partners of very general Hodge-special Gushel-Mukai fourfolds with associated K3 surface, and a lower bound for this number in the case of a twisted associated K3 surface.
We study a certain class of degenerations of Gushel-Mukai fourfolds as conic bundles, which we call tame degenerations and which are natural if one wants to prove that very general Gushel-Mukai fourfolds are irrational using the degeneration method due to Voisin, Colliot-Thélène-Pirutka, Totaro et al. However, we prove that no such tame degenerations do exist.
We show that the stability conditions on the Kuznetsov component of a Gushel-Mukai threefold, constructed by Bayer, Lahoz, Macrì and Stellari, are preserved by the Serre functor, up to the action of the universal cover of $\text{GL}^+_2(\mathbb{R})$. As application, we construct stability conditions on the Kuznetsov component of special Gushel-Mukai fourfolds.
We describe the moduli stack of Gushel-Mukai varieties as a global quotient stack and its coarse moduli space as the corresponding GIT quotient. The construction is based on a comprehensive study of the relation between this stack and the stack of Lagrangian data; roughly speaking, we show that the former is a generalized root stack of the latter. As an application, we define the period map for Gushel-Mukai varieties and construct some complete nonisotrivial families of smooth Gushel-Mukai varieties. In an appendix, we describe a generalization of the root stack construction used in our approa…
We present some applications of the Macaulay2 software package SpecialFanoFourfolds, a package for working with Hodge-special cubic fourfolds and Hodge-special Gushel--Mukai fourfolds. In particular, we show how to construct new examples of such fourfolds, some of which turn out to be rational. We also describe how to calculate K3 surfaces associated with cubic or Gushel-Mukai fourfolds, which relies on an explicit unirationality of some moduli spaces of K3 surfaces.
We give explicit descriptions of some Noether-Lefschetz divisors in the moduli space of Gushel-Mukai fourfolds. As a consequence we obtain that their Kodaira dimension is negative.
We show that Gushel-Mukai fivefolds admit a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. As a consequence, a certain tautological subring of the Chow ring of powers of these varieties injects into cohomology.
We prove that the invariant locus of the involution associated to a general double EPW sextic is a constant surface and introduce a filtration on $\CH_1$ of a Gushel-Mukai fourfold. We verify the sheaf/cycle correspondence for sheaves supported on low degree rational curves, parallel to the cubic fourfolds case.
We perform a systematic study of Gushel-Mukai varieties---quadratic sections of linear sections of cones over the Grassmannian Gr(2,5). This class of varieties includes Clifford general curves of genus 6, Brill-Noether general polarized K3 surfaces of genus 6, prime Fano threefolds of genus 6, and their higher-dimensional analogues. We establish an intrinsic characterization of normal Gushel-Mukai varieties in terms of their excess conormal sheaves, which leads to a new proof of the classification theorem of Gushel and Mukai. We give a description of isomorphism classes of Gushel-Mukai varieti…
In analogy to the case of cubic fourfolds, we discuss the conditions under which the double cover $\tilde{Y}_A$ of the EPW sextic hypersurface associated to a Gushel-Mukai fourfold is birationally equivalent to a moduli space of (twisted) stable sheaves on a K3 surface. In particular, we prove that $\tilde{Y}_A$ is birational to the Hilbert scheme of two points on a K3 surface if and only if the Gushel-Mukai fourfold is Hodge-special with discriminant $d$ such that the negative Pell equation $\mathcal{P}_{d/2}(-1)$ is solvable in $\mathbb{Z}$.
We study the Hilbert scheme $\mathcal{H}$ of twisted cubics on a special smooth Gushel-Mukai threefolds $X_{10}$. We show that it is a smooth irreducible projective threefold if $X_{10}$ is general among special Gushel-Mukai threefolds, while it is singular if $X_{10}$ is not general. We construct an irreducible component of a moduli space of Bridgeland stable objects in the Kuznetsov component of $X_{10}$ as a divisorial contraction of $\mathcal{H}$. We also identify the minimal model of Fano surface $\mathcal{C}(X_{10}')$ of conics on a smooth ordinary Gushel-Mukai threefold with moduli spac…
We study the derived categories of coherent sheaves on Gushel-Mukai varieties. In the derived category of such a variety, we isolate a special semiorthogonal component, which is a K3 or Enriques category according to whether the dimension of the variety is even or odd. We analyze the basic properties of this category using Hochschild homology, Hochschild cohomology, and the Grothendieck group. We study the K3 category of a Gushel-Mukai fourfold in more detail. Namely, we show that this category is equivalent to the derived category of a K3 surface for a certain codimension 1 family of rational…
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…
Northwestern University（西北大学）；Shanghai Center for Mathematical Sciences, Fudan University（复旦大学上海数学中心）；Department of Mathematics, Nanjing University（南京大学数学系）；Kaiyuan International Mathematical Sciences Institute（开普勒国际数学科学研究所）
The object of this note is the moduli spaces of cubic fourfolds (resp., Gushel-Mukai fourfolds) which contain some special rational surfaces. Under some hypotheses on the families of such surfaces, we develop a general method to show the unirationality of the moduli spaces of the $n$-pointed such fourfolds. We apply this to some codimension 1 loci of cubic fourfolds (resp., Gushel-Mukai fourfolds) appeared in the literature recently.
We construct new families of smooth Fano fourfolds with Picard rank $1$ which contain open $\Bbb A^1$-cylinders, that is, Zariski open subsets of the form $Z \times \Bbb A^1$, where $Z$ is a quasiprojective variety. In particular, we show that every Mukai fourfold of genus $8$ is cylindrical and there exists a family of cylindrical Gushel-Mukai fourfolds.
We study Fourier-Mukai transforms for smooth projective varieties whose canonical bundles have finite order, and relate them to equivariant transforms on certain finite covering spaces. Our results lead to new equivalences of derived categories for Enriques and bielliptic surfaces.
This note is an answer to a problem proposed by Ranestad and Iliev. We prove that the projection of general nodal linear sections of suitable dimension of the Mukai varieties $M_g$ are linear sections of $M_{g-1}$.
We study the Fourier-Mukai functor D(Y) -> D(X) induced by the universal family on a fine moduli space Y for simple semihomogeneous vector bundles on an abelian variety X. The main result is that the Fourier-Mukai transform of a very negative line bundle on Y is ample if and only if the bundles parametrized by Y are nef.
Mukai varieties are Fano varieties of Picard number one and coindex three. In genus seven to ten they are linear sections of some special homogeneous varieties. We describe the generic automorphism groups of these varieties. When they are expected to be trivial for dimensional reasons, we show they are indeed trivial, up to three interesting and unexpected exceptions in genera 7, 8, 9, and codimension 4, 3, 2 respectively. We conclude in particular that a generic prime Fano threefold of genus g has no automorphisms for 7 $\le$ g $\le$ 10. In the Appendix by Y. Prokhorov, the latter statement i…
We associate to any complete spherical variety $X$ a certain nonnegative rational number $\wp(X)$, which we conjecture to satisfy the inequality $\wp(X) \le \operatorname{dim} X - \operatorname{rank} X$ with equality holding if and only if $X$ is isomorphic to a toric variety. We show that, for spherical varieties, our conjecture implies the generalized Mukai conjecture on the pseudo-index of smooth Fano varieties due to Bonavero, Casagrande, Debarre, and Druel. We also deduce from our conjecture a smoothness criterion for spherical varieties. It follows from the work of Pasquier that our conj…
Let X be a Fano variety of dimension n, pseudoindex i_X and Picard number ρ_X. A generalization of a conjecture of Mukai says that ρ_X(i_X-1)\le n. We prove that the conjecture holds if: a) X has pseudoindex i_X \ge \frac{n+3}{3} and either has a fiber type extremal contraction or does not have small extremal contractions b) X has dimension five.
We discuss birational properties of Mukai varieties, i.e., of higher-dimensional analogues of prime Fano threefolds of genus $g \in \{7,8,9,10\}$ over an arbitrary field $\mathsf{k}$ of zero characteristic. In the case of dimension $n \ge 4$ we prove that these varieties are $\mathsf{k}$-rational if and only if they have a $\mathsf{k}$-point except for the case of genus $9$, where we assume $n \ge 5$. Furthermore, we prove that Mukai varieties of genus $g \in \{7,8,9,10\}$ and dimension $n \ge 5$ contain cylinders if they have a $\mathsf{k}$-point. Finally, we prove that the embedding $X \hook…
We will discuss the Fourier-Mukai partners of a given abelian variety. The first part of the note is to give some basic theory of Fourier-Mukai partners and semi-homogenous vector bundles, then we will discuss the case when the kernel of an equivalence is given by a semi-homogenous vector bundle. In particular, given an abelian variety B, and a simple semi-homogenous vector bundle E on B, we will discuss for which E, it can be the image of the structure sheaf of the unit on A under some triangulated equivalence, where A is some abelian variety.
This article is based on a talk given at the Kinosaki Symposium on Algebraic Geometry in 2015, about a work in progress. We describe a polarization on a derived equivalent abelian variety by using Fourier-Mukai theory. We explicitly formulate a conjecture which says certain Fourier-Mukai transforms between derived categories give equivalences of some hearts of Bridgeland stability conditions. We establish it for abelian surfaces, which is already known due to D. Huybrechts, and for abelian 3-folds. This generalizes the author's previous joint work with A. Maciocia on principally polarized abel…
We study the Fourier-Mukai transform for holonomic D-modules on complex abelian varieties. Among other things, we show that the cohomology support loci of a holonomic D-module are finite unions of linear subvarieties, which go through points of finite order for objects of geometric origin; that the standard t-structure on the derived category of holonomic complexes corresponds, under the Fourier-Mukai transform, to a certain perverse coherent t-structure in the sense of Kashiwara and Arinkin-Bezrukavnikov; and that Fourier-Mukai transforms of simple holonomic D-modules are intersection complex…
A Mukai variety is a Fano n-fold of index n-2. In this paper we study the fundamental divisor of a Mukai variety with at worst log terminal singularities. The main result is a complete classification of log terminal Mukai varieties which have not good divisors, examples of "bad" varieties are given. In such a way we also give a shorter proof of Mukai Conjecture, solved in our previous paper alg-geom/9611024.
