Analytic K-semistability and local wall-crossing

TL;DR

This paper establishes a link between K-polystability and the existence of constant scalar curvature K"ahler metrics for small deformations, revealing local wall-crossing phenomena in moduli spaces.

Contribution

It proves that K-polystability implies the existence of cscK metrics under certain conditions and reduces stability checks to Futaki invariant computations.

Findings

K-polystability implies existence of cscK metrics in small deformations.

Reduction of K-polystability to Futaki invariant calculations.

Identification of local wall-crossing phenomena in moduli of cscK manifolds.

Abstract

For a small polarised deformation of a constant scalar curvature K\"ahler manifold, under some cohomological vanishing conditions, we prove that K-polystability along nearby polarisations implies the existence of a constant scalar curvature K\"ahler metric. In this setting, we reduce K-polystability to the computation of the classical Futaki invariant on the cscK degeneration. Our result holds on specific families and provides local wall-crossing phenomena for the moduli of cscK manifolds when the polarisation varies.