Kollar—Shepherd-Barron—Alexeev (KSBA) have given a general construction that provides a geometric compactification for the moduli space of varieties of general type. Unfortunately, even in relatively simple cases (e.g. surfaces of general type with small invariants) it is difficult to understand the boundary points and the structure of this KSBA compactification. Thus, it is natural to try to compare the KSBA construction with other constructions, in particular with Hodge theoretic constructions of the moduli space. The Hodge theoretic construction has the advantage of having a lot of structure (of arithmetic and representation theoretic nature), but except a few cases (essentially abelian varieties and K3s) it is highly transcendental. In this talk, I will report on joint work with P. Griffiths, M. Green and C. Robles on the study of the moduli and periods of H-surfaces (Horikawa surfaces). The H-surfaces are surfaces of general type with p_g=2, q=0, K^2=2. They are essentially the simplest case where both the KSBA and Hodge theoretic construction are non-trivial. Considering and comparing the two approaches gives a rich picture which suggests an important role for the period map in the study of moduli spaces beyond the classical cases of abelian varieties and K3s.
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