Local existence of solutions to polynomial curvature equations is equivalent to local polystability, with local filtrations and a Kempf-Ness homeomorphism.
Analytic K-semistability and local wall-crossing
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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.
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Perturbations of Vector Bundle whose Curvature Form Solves a Polynomial Equation
Local existence of solutions to polynomial curvature equations is equivalent to local polystability, with local filtrations and a Kempf-Ness homeomorphism.