Degeneration of Hodge structures on I-surfaces

TL;DR

This paper investigates the Hodge theoretic properties of I-surfaces, demonstrating that all predicted degenerations in their moduli space are geometrically realized, thus advancing understanding of their Hodge structure degenerations.

Contribution

It computes the Hodge types of known I-surfaces with specific invariants and confirms that all predicted degenerations occur in practice.

Findings

All predicted degenerations are geometrically realized.

Hodge types of known I-surfaces are explicitly computed.

Supports the stratification of the moduli space via Hodge data.

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.