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

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.

43 pages. v.3: more details added, proof of Theorem 3.9 revised, typos corrected and other minor revisions