Prescribed mean curvature problems on homogeneous vector bundles

TL;DR

This paper develops a new algebraic criterion for the existence of singular Hermitian structures with prescribed mean curvature on homogeneous vector bundles, reducing complex problems to abelian theory and overcoming classical curvature restrictions.

Contribution

It introduces an explicit algebraic splitting criterion for homogeneous bundles, enabling the reduction of non-abelian problems to abelian line bundle theory and constructing singular structures with prescribed singularities.

Findings

Established an algebraic criterion for bundle splitting.

Reduced the prescribed mean curvature problem to abelian line bundle theory.

Provided conditions for realizing functions as mean curvature of singular structures.

Abstract

In this paper, we investigate the existence of weak singular Hermite-Einstein structures on homogeneous holomorphic vector bundles over rational homogeneous varieties. Using Cartan's highest weight theory, we establish an explicit algebraic criterion for a homogeneous vector bundle ${\bf{E}}$ to admit a topological splitting ${\bf{E}} \cong {\bf{E}}_{0} \otimes {\bf{L}}_{0}$, where ${\bf{L}}_{0} \in {\rm{Pic}}(X)$ and $c_{1}({\bf{E}}_{0}) = 0$. When this condition is satisfied, the prescribed mean curvature equation completely decouples. By shifting the topological obstruction entirely to the line bundle ${\bf{L}}_{0}$, this splitting reduces the non-abelian prescribed mean curvature problem on ${\bf{E}}$ to Demailly's abelian theory of singular line bundle metrics. As a main application, we obtain a sufficient algebraic condition, expressed in terms of intersection numbers, under which an $L^{2}$-function can be realized as the mean curvature of a singular Hermitian structure on an irreducible homogeneous bundle. Ultimately, by overcoming the bounded curvature restrictions inherent to the classical Bando-Siu framework, this approach provides a robust mechanism to construct singular Hermitian structures accommodating prescribed singularities along analytic subvarieties.