Hermitian-Yang-Mills connections on some complete non-compact Kähler manifolds (Q6624843)

A projective Hermitian-Einstein (PHE) metric on a holomorphic vector bundle \(E\) with respect to a Kähler form \(\omega\) is one such that \(F_H \wedge \omega^{n-1}=\frac{tr_{\omega}(F_H)}{r} \omega^n\). The Donaldson-Uhlenbeck-Yau theorem guarantees their existence iff slope stability holds for compact manifolds. In the noncompact case, considerable amount of work was done (notably by Mochizuki and Simpson among others). However, the stability condition is analytic. In this paper, the noncompact manifold \(X\) is the complement of a divisor \(D\) in a compact one \(\bar{X}\). Likewise, the vector bundle \(E\) is the restriction from a bundle on \(\bar{X}\). This paper needs the Kähler metric \(\omega\) to behave in a particular manner near \(D\) (Assumption 1). It turns out that Calabi-Yau metrics on the complement of anticanonical divisors on Fano manifolds (which exist thanks to Tian-Yau, and more generally, due to Hein-Sun-Viaclosky-Zhang) satisfy this property and hence the theorem in this paper can be applied to an interesting case. The main theorem (Theorem 1.3) is a Kobayashi-Hitchin-type correspondence: Suppose \(\omega=\omega_0+\sqrt{-1}\partial \bar{\partial} \phi\) satisfies Assumption 1 (where \(\omega_0\) is smooth on \(\bar{X}\) but vanishes on the divisor). Assume that \(\bar{E}\vert_{D}\) is \(c_1(N_D)\)-polystable. Then there exists a unique \(\omega\)-PHE metric in a certain class of metrics (called \(\mathcal{P}_{H_0})\) iff \(\bar{E}\) is \((c_1(D),[\omega_0])\)-polystable (a notion of stability using a pair is defined in this paper). The proof (of the ``hard'' direction) is by constructing solutions to a Dirichlet problem on an exhaustion (using an existence result of Donaldson) and then proving estimates. As always, the zeroeth-order estimate is the hardest and a key result of Uhlenbeck and Yau on weakly holomorphic subbundles (adapted to this noncompact setting) is used for the same. In addition, one needs weighted Sobolev inequalities too.