Limiting mixed Hodge structures associated to I-surfaces with simple elliptic singularities

TL;DR

This paper investigates the asymptotic behavior of the period map for I-surfaces with simple elliptic singularities, analyzing their mixed Hodge structures, monodromy, and establishing a global Torelli theorem for certain boundary strata.

Contribution

It provides a detailed analysis of the limiting mixed Hodge structures and monodromy for I-surfaces with elliptic singularities, and proves a global Torelli theorem for specific boundary cases.

Findings

Identified 6 boundary strata with specific Hodge structure properties

Determined that the nilpotent orbit uniquely determines the boundary stratum

Proved a global Torelli theorem for one boundary stratum

Abstract

An I-surface $X$ is a surface of general type with $K_X^2 =1$ and $p_g(X) =2$. This paper studies the asymptotic behavior of the period map for I-surfaces acquiring simple elliptic singularities. First we describe the relationship between the deformation theory of such surfaces and their $d$-semistable models. Next we analyze the mixed Hodge structures on the $d$-semistable models, the corresponding limiting mixed Hodge structures, and the monodromy. There are $6$ possible boundary strata for which the relevant limiting mixed Hodge structures satisfy: $\dim W_1 = 4$, and hence $W_2/W_1$ is of pure type $(1,1)$. We show that, in each case, the nilpotent orbit of limiting mixed Hodge structures determines the boundary stratum and prove a global Torelli theorem for one such stratum.