Degeneration of Hodge structures on I-surfaces

Abstract

Work of Green, Griffiths, Laza, and Robles suggests that the moduli space of (smoothable) stable surfaces should admit a natural stratification defined via Hodge theoretic data. In the case of stable surfaces with $K_X^2=1$ and $\chi (X)=3$ we compute the Hodge type of all examples known to us and show that all predicted degenerations are geometrically realised.