The GIT stability and Hodge structures of hypersurfaces via minimal exponent

Abstract

Let $X\subset \mathbb P^n$ be a degree $d$ hypersurface. We prove that $X$ is GIT stable if the minimal exponent $\widetilde \alpha(X)>\frac{n+1}{d}$ and GIT semistable if $\widetilde \alpha(X)=\frac{n+1}{d}$, resolving a question of Laza. Conversely, for GIT semistable cubic hypersurfaces, we prove a uniform lower bound for the minimal exponent, which implies that every such cubic has canonical singularities (and is terminal for $n\ge 6$), answering a question of Spotti-Sun. In the classical cases $(n,d)=(2,4),(2,6),(3,3),(4,3),(5,3)$, the period map from the GIT moduli is an open embedding over the stable locus with $\widetilde \alpha(X)>\frac{n+1}{d}$ and extends regularly to the Baily-Borel compactification precisely along the boundary where $\widetilde \alpha(X)=\frac{n+1}{d}$. To generalize this period map behavior in the Calabi-Yau type case $\frac{n+1}{d}=m+1\in \mathbb Z$, we introduce $m$-liminal sources and $m$-liminal centers, refining the theory of sources and log canonical centers. For an $m$-Du Bois hypersurface, we prove that the core of the limit mixed Hodge structure of any one-parameter smoothing is completely determined by the $m$-liminal source. In particular, maximal unipotent degeneration is detected by the local singularity type of the special fiber.