### Working paper

## On the cohomology of Gushel-Mukai sixfolds

This paper performs a systematic study of Gushel–Mukai varieties —Fano mani- folds with Picard number 1, coindex 3, and degree 10 (higher-dimens ional analogues of prime Fano threefolds of genus 6). We introduce a new approach to the c lassification of these va- rieties which includes mildly singular varieties, gives a criterion for an iso morphism of such varieties, and describes their automorphisms groups. We carefully develop the relation between Gushel–Mukai varieties an d Eisenbud–Popescu– Walter sextics introduced earlier by Iliev–Manivel and O’Grady. We de scribe explicitly all Gushel–Mukai varieties whose associated EPW sextics are isomorph ic or dual (we call them period partners or dual varieties respectively). Finally, we show th at in dimension 3 and higher, period partners/dual varieties are always birationally isomo rphic.

Beauville and Donagi proved in 1985 that the primitive middle cohomology of a smooth complex cubic 4-fold and the primitive second cohomology of its variety of lines, a smooth hyper-Kähler 4-fold, are isomorphic as polarized integral Hodge structures. We prove analogous statements for smooth complex Gushel–Mukai varieties of dimension 4 (resp., 6), that is, 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 hyper-Kähler 4-fold is in both cases a smooth double cover of a hypersurface in P^5 called an *Eisenbud–Popescu–Walter sextic.*

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.