10 problems

A holomorphic vector bundle over a closed Kähler manifold is called semistable when it satisfies the usual slope-semistability condition. An approximate Hermitian–Einstein structur…

Kobayashi–Hitchin conjecture. The equivalence between stability and the existence of a Hermitian–Einstein structure should hold for holomorphic vector bundles over compact Kähler m…

The polystability criterion. There exists a $Z$-positive solution $h\in\mathcal{H}^+(E)$ of the $Z$-critical equation if and only if $E=\bigoplus_i E_i$ is $Z$-polystable and every…

The $Z$-critical correspondence. There exists a $Z$-positive solution of the $Z$-critical equation if and only if $E$ is $Z$-stable, where $Z$-stability means that for every cohere…

dHYM–Bridgeland stability conjecture. The existence of a dHYM instanton $\nabla$ on ${\bf E}$ should be equivalent to Bridgeland stability of ${\bf E}$. This conjecture proposes an…

Let $(E,\bar{\partial}_{A})$ be a holomorphic vector bundle, and let $Gr^{HNS}(E,\bar{\partial}_{A})$ denote the associated graded object of its Harder–Narasimhan–Seshadri filtrati…

Let $M$ be a compact complex manifold, let $E$ be a holomorphic vector bundle over $M$, and let $\Gamma\subset M\times M$ be a transversal correspondence. Let…

Let $M$ be a compact, connected Riemann surface, let $E\rightarrow M$ be a holomorphic vector bundle with a smooth Hermitian metric $h$, and let $dV$ be a smooth volume form. For a…

Let $M$ be an algebraic manifold with $\dim_{\mathbb{C}}M\geq3$, let $L$ be a very ample line bundle on $M$, and let $\omega\in K(M,L,x\,dy-y\,dx)$ be a foliation with normal bundl…

Let $X$ be a compact Kähler manifold and let $E$ be a holomorphic vector bundle over $X$. Stability is understood in the algebro-geometric sense, and a unitary Hermitian–Yang–Mills…