Analytic K-semistability and local wall-crossing

Abstract

For a small polarised deformation of a constant scalar curvature Kähler manifold, under some cohomological vanishing conditions, we prove that K-polystability along nearby polarisations implies the existence of a constant scalar curvature Kähler 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.

A gap in the proof of former Proposition 3.3 (perturbation of the Kuranishi slice) forced us to use a new approach relying on Hodge theory for Dolbeault cohomology on manifolds with boundary, and moment map flow arguments. This results in slightly different assumptions. To appear in AGAG