Images of extended period maps

As a geometric application of polarized log Hodge structures, we show the following. Let $M_{H}^{\mathrm {sm}}$ be a projective variety which is a compactification of the coarse moduli space of surfaces of general type constructed by Kawamata, Kollár, Shepherd-Barron, Alexeev, Mori, Karu, et al., and let $\Gamma \backslash D_{\Sigma }$ be a log manifold which is the fine moduli space of polarized log Hodge structures constructed by Kato and Usui. If we take a suitable finite cover $M’\to M_{i}$ of any irreducible component $M_{i}$ of $M_{H}^{\mathrm {sm}}$, and if we assume the existence of a suitable fan $\Sigma$, then there is an extended period map $\psi :M’\to \Gamma \backslash D_{\Sigma }$ and its image is the analytic subspace associated to a separated compact algebraic space. The point is that, although $\Gamma \backslash D_{\Sigma }$ is a “log manifold” with slits, the image $\psi (M’)$ is not affected by these slits and is a classical familiar object: a separated compact algebraic space.