The equivalence classes of smooth algebraic varieties X of a particular type form its moduli space M , and their study is a central problem in algebraic geometry. When X is of general type M exists and has a canonical compactification M as a projective algebraic variety. Aside from a few classical cases (curves, K3 surfaces, abelian varieties) very little is known about the boundary ∂M=M̅ \M and the singular varieties X0 that corresponds to boundary points. In this talk we will explain how Hodge theory provides basic invariants of the X0 's and in some early examples may be used to help understand geometrically the boundary structure of moduli.
