The coupled equations imply alpha-K-polystability for triples

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. The coupled equations imply alpha-K-polystability for triples

   Let $(X,L,E)$ be a smooth polarized variety and a holomorphic vector bundle over $X$ of rank $r$ and degree zero, where $L$ is ample. Let $\omega \in c_1(L)$ be a Kähler metric and let $H$ be a Hermitian metric on $E$, with coupling constants $\alpha_0,\alpha_1$ and $\alpha=\frac{r\pi^2\alpha_1}{\alpha_0}$. The coupled-equations conjecture. If $\omega$ and $H$ satisfy the coupled equations, then the triple $(X,L,E)$ is $\alpha$-K-polystable. This conjecture proposes an algebro-geometric obstruction to solutions of the coupled Kähler and Yang–Mills equations; the text gives supporting evidence from vanishing and nonnegativity of the associated invariant for product and base-preserving test configurations.

   source: Mario Garcia-Fernandez, “Coupled equations for Kähler metrics and Yang-Mills connections (Thesis)”, arXiv:1102.0985 (2011).