Gushel–Mukai varieties: Moduli

Abstract

Gushel-Mukai (or GM for short) varieties are smooth (complex) dimensionally transverse intersections of a cone over the Grassmannian Gr(2, 5) with a linear space and a quadratic hypersurface. They occur in each dimension 1 through 6 and they are Fano varieties (their anticanonical bundle is ample) in dimensions 3, 4, 5, and 6. The aim of this series of lectures 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, and O'Grady.These varieties first appeared in the classification of complex Fano threefolds: Gushel showed that any smooth prime Fano threefold of degree 10 is a GM variety. Mukai later extended Gushel's results and proved that all Fano varieties of coindex 3, degree 10, and Picard number 1, all Brill-Noether-general polarized K3 surfaces of degree 10, and all Clifford-general curves of genus 6 are GM varieties.Why are people interested in these varieties? One of the reasons why their geometry is so rich is that, to any GM variety is canonically attached a sextic hypersurface in P 5 , called an Eisenbud-Popescu-Walter (or EPW for short) sextic and a canonical double cover thereof, a hyperkähler fourfold called a double EPW sextic. In some sense, the pair consisting of a GM variety and its double EPW sextic behaves very much (but with more complications, and also some differences) like a cubic hypersurface in P 5 and its (hyperkähler) variety of lines. Not many examples of these pairs-a Fano variety and a hyperkähler variety-are known (another example are the hyperplane sections of the Grassmannian Gr(3, 10) and their associated Debarre-Voisin hyperkähler fourfold), so it is worth looking in depth into one of these.The first main difference is that there is a positive-dimensional family of GM varieties attached to the same EPW sextic; this family can be precisely described. Another distinguishing feature is that each EPW sextic has a dual EPW sextic (its projective dual). GM varieties associated with isomorphic or dual EPW sextics are all birationally isomorphic.A common feature with cubic fourfolds is that the middle Hodge structure of a GM variety of even dimension is isomorphic (up to a Tate twist) to the second cohomology of its associated double EPW sextic. Together with the Verbitsky-Torelli theorem for hyperkähler fourfolds, this leads to a complete description of the period map for even-dimensional GM varieties.In odd dimensions, say 3 or 5, a GM variety has a 10-dimensional intermediate Jacobian and we show that it is isomorphic to the Albanese variety of a surface of general type canonically attached to the EPW sextic.Another aspect of GM varieties that makes them very close to cubic fourfolds is the rationality problem. Whereas (most) GM varieties of dimensions at most 3 are not rational, and GM varieties of dimensions at least 5 are all rational, the situation for GM fourfolds is still mysterious: some rational examples are known but, although one expect...

Derived categories of Gushel–Mukai varieties

“…Theorem 3.1 suggests there is a morphism from the moduli stack of -dimensional GM varieties (see Appendix A) to the quotient stack (where is the Lagrangian Grassmannian) given by at the level of points, whose fiber over a point is the union of two EPW strata , modulo the action of the finite stabilizer group of in . This morphism will be discussed in detail in [DK18b]. Let us simply note that it gives a geometric way to compute (cf.…”

“…The following result gives the basic properties of the moduli stack . An explicit description of will be given in [DK18b]. We follow [Sta17] for our conventions on algebraic stacks.…”

Some New Rational Gushel Fourfolds

“…In both cases, the map γ X from X to G(1, 4) is called the Gushel map. By results proved in [3] (see also [4][5][6]), Fano fourfolds as above are parameterized (up to isomorphism) by the points of a coarse moduli space M 4 of dimension 24, where the Gushel fourfolds correspond to the points of a closed irreducible subvariety M G 4 ⊂ M 4 of codimension 2. A fourfold [X] ∈ M 4 is said to be special (or Hodge-special ) if it contains a surface whose cohomology class does not lie in γ * X (H 4 (G(1, 4), Z)); equivalently, [X] is special if and only if rk(H 2,2 (X) ∩ H 4…”

Gushel--Mukai varieties: intermediate Jacobians

“…This article is an addition to the series [DK18, DK19, DK20b, KP18] on the geometry of Gushel-Mukai varieties. For an introduction, we recommend the survey [Deb20].…”