Moduli spaces of polarised K3 surfaces and irreducible symplectic manifolds

Moduli spaces of polarized K3 surfaces and irreducible symplectic manifolds can be related to quotients of homogeneous domains of type IV by arithmetic groups. The latter quotients can then be studied using quasi-pullbacks of the Borcherds form.
In this talk we will discuss the series consisting of K3 surfaces, irreducible symplectic manifolds of K3^[n] type and O'Grady's 10-dimensional examples. The dimensions of the domains in question are 19, 20 and 21. Although the approach is similar in all three cases, there are also major differences. In particular the last case is special due to the special geometry of roots in this case. This makes it possible to treat this case without using either sophisticated analytic number theory or the help of a computer. This is part of an ongoing joint project with V.Gritsenko and G.K.Sankaran.