Valery Gritsenko

The Kodaira dimension of the moduli of K3 surfaces

The global Torelli theorem for projective K3 surfaces was first proved by Piatetskii-Shapiro and Shafarevich 35 years ago, opening the way to treating moduli problems for K3 surfaces. The moduli space of polarised K3 surfaces of degree 2d is a quasi-projective variety of dimension 19. For general d very little has been known hitherto about the Kodaira dimension of these varieties. In this paper we present an almost complete solution to this problem. Our main result says that this moduli space is of general type for d>61 and for d=46, 50, 54, 57, 58, 60.

Moduli spaces of irreducible symplectic manifolds

We study the moduli spaces of polarised irreducible symplectic manifolds. By a comparison with locally symmetric varieties of orthogonal type of dimension 20, we show that the moduli space of polarised deformation K3 manifolds with polarisation of degree 2d and split type is of general type if d ≥ 12. MSC 2000: 14J15, 14J35, 32J27, 11E25, 11F55

Siegel modular forms of genus 2 with the simplest divisor

We prove that there exist exactly eight Siegel modular forms with respect to the congruence subgroups of Hecke type of the paramodular groups of genus 2 vanishing precisely along the diagonal of the Siegel upper half-plane. This is a solution of a question formulated during the conference ‘Black holes, Black Rings and Modular Forms’ (ENS, Paris, August 2007). These modular forms generalize the classical Igusa form and the forms constructed by Gritsenko and Nikulin in 1998.

On classification of Lorentzian Kac-Moody algebras

The general theory of Lorentzian Kac-Moody algebras is considered. This theory must serve as a hyperbolic analogue of the classical theories of finite-dimensional semisimple Lie algebras and affine Kac-Moody algebras. The first examples of Lorentzian Kac-Moody algebras were found by Borcherds. Here general finiteness results for the set of Lorentzian Kac-Moody algebras of rank ≥3 are considered along with the classification problem for these algebras. As an example, a classification is given for Lorentzian Kac-Moody algebras of rank 3 with hyperbolic root lattice , symmetry lattice , and symmetry group , , where and are given by and is trivial on , is the extended paramodular group. This is perhaps the first example in which a large class of Lorentzian Kac-Moody algebras has been classified.

Uniruledness of orthogonal modular varieties

AbstractA strongly reflective modular form with respect to an orthogonalgroup of signature (2,n) determines a Lorentzian Kac–Moody alge-bra. We find a new geometric application of such modular forms: weprove that if the weight is larger than n then the corresponding modu-lar variety is uniruled. We also construct new reflective modular formsand thus provide new examples of uniruled moduli spaces of lattice po-larised K3 surfaces. Finally we prove that the moduli space of Kummersurfaces associated to (1,21)-polarised abelian surfaces is uniruled. 1 Reflective modular forms Let L be an even integral lattice with a quadratic form of signature (2,n)and letD(L) = {[Z] ∈ P(L⊗C) | (Z,Z) = 0, (Z,Z) > 0} + be the associated n-dimensional bounded symmetric Hermitian domain oftype IV (here + denotes one of its two connected components). We denoteby O + (L) the index 2 subgroup of the integral orthogonal group O(L) pre-serving D(L). For any v ∈ L ⊗ Qsuch that v 2 = (v,v) < 0 we define therational quadratic divisorD

Lorentzian Kac-Moody algebras with Weyl groups of 2-reflections

We describe a new large class of Lorentzian Kac--Moody algebras. For all ranks, we classify 2-reflective hyperbolic lattices S with the group of 2-reflections of finite volume and with a lattice Weyl vector. They define the corresponding hyperbolic Kac--Moody algebras of restricted arithmetic type which are graded by S. For most of them, we construct Lorentzian Kac--Moody algebras which give their automorphic corrections: they are graded by the S, have the same simple real roots, but their denominator identities are given by automorphic forms with 2-reflective divisors. We give exact constructions of these automorphic forms as Borcherds products and, in some cases, as additive Jacobi liftings.