The GIT stability and Hodge structures of hypersurfaces via minimal exponent

TL;DR

This work develops a unified framework linking GIT stability of degree $d$ hypersurfaces in $\mathbb{P}^n$ with Hodge-theoretic degeneration data via the minimal exponent $\widetilde{\alpha}$. It proves sharp stability criteria: $X$ is GIT stable if $\widetilde{\alpha}(X) > \frac{n+1}{d}$ and semistable when equal, and connects these to the affine-cone Hilbert–Mumford bounds. For cubic hypersurfaces, it yields uniform lower bounds on $\widetilde{\alpha}$, giving canonical (and terminal for $n\ge6$) singularities for semistable cubics, answering questions of Spotti–Sun and related works. The paper then builds a Calabi–Yau-type theory of $m$-liminal sources/centers to describe limit mixed Hodge structures in degenerations, showing the core of the middle cohomology is determined by these liminal data and that maximal degeneration is detected by local singularity types. Finally, it extends period-map perspectives to conjectural Baily–Borel-type compactifications for Calabi–Yau-type hypersurfaces and provides Thom–Sebastiani-type results for liminal sources, with explicit examples across dimensions.

Abstract

Let $X\subset \mathbb P^n$ be a degree $d$ hypersurface. We prove that $X$ is GIT stable if the minimal exponent $\widetilde α(X)>\frac{n+1}{d}$ and GIT semistable if $\widetilde α(X)=\frac{n+1}{d}$, 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 $\widetilde α(X)>\frac{n+1}{d}$ and extends regularly to the Baily-Borel compactification precisely along the boundary where $\widetilde α(X)=\frac{n+1}{d}$. 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.

Theorems & Definitions (88)

• Theorem 1
• Theorem 2
• Corollary 3: Shah80Kondo00ACT02LS07Artebani09Looijenga09Laza10ACT11
• Conjecture 4
• Theorem 5
• Theorem 6
• Corollary 7
• Theorem 8
• Definition 1.1
• Proposition 1.2