Hodge theory and K-stability of some very symmetric hypersurfaces

TL;DR

This paper investigates the Hodge structures and K-stability of symmetric hypersurfaces arising from the moduli space of hypersurfaces, including explicit computations and stability results for certain degenerate cases.

Contribution

It provides explicit Hodge and intersection cohomology calculations and establishes K-polystability for specific symmetric hypersurfaces, connecting geometric stability with Hodge theory.

Findings

Computed Hodge structures on singular and intersection cohomology.

Proved K-polystability of certain mildly singular hypersurfaces.

Identified a class of hypersurfaces that are K-polystable for l ≥ 2.

Abstract

We study some interesting hypersurfaces that naturally arise when studying the period map on the moduli space of hypersurfaces, in the context of Sung Gi Park's recent work on studying the GIT moduli space of hypersurfaces via the minimal exponent. We compute the Hodge structure on the singular cohomology and the intersection cohomology of these hypersurfaces, and also show the $K$-polystability of certain mildly singular degenerate hypersurfaces among them. In particular, the following hypersurface is $K$-polystable for $l \geq 2$: $$ \{ x_{11}\cdots x_{1d} + \ldots + x_{ld} \cdots x_{ld} = 0\} \subset \PP^{ld-1}.$$