Local existence of solutions to polynomial curvature equations is equivalent to local polystability, with local filtrations and a Kempf-Ness homeomorphism.
Semi-stability and local wall-crossing for hermitian Yang-Mills connections
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abstract
We consider a sufficiently smooth semi-stable holomorphic vector bundle over a compact K\"ahler manifold. Assuming the automorphism group of its graded object to be abelian, we provide a semialgebraic decomposition of a neighbourhood of the polarisation in the K\"ahler cone into chambers characterising (in)stability. For a path in a stable chamber converging to the initial polarisation, we show that the associated HYM connections converge to an HYM connection on the graded object.
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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.