The moduli spaces of polarised symplectic varieties

The first example of such varieties is a polarised K3 surface. In 1956 A. Weil formulated a program on the K3 surfaces an their moduli spaces. All questions of the program had been solved during the next 25 years except the problem on the birational type of the moduli space $F(2d)$ of the polarized K3 surfaces of degree $2d$. ($F(2d)$ is quasi-projective variety of dimension 19). For $d=1,2,\dots,10,12,17,19$ the variety $F(2d)$ is still unirational (Mukai). In this talk I present m joint result with K. Hulek (Hannover) and G. Sankaran (Bath) on the solution of the problem of the birational type. Using the theory of automorphic forms we proved that $F(2d)$ is of general type, i.e., its Kodaira dimension is maximal, starting from $d=46$. Our method gives similar results for the moduli spaces (of dimension 20 and 21) of some polarised irreducible symplectic manifolds.