Generalized Donaldson's functionals and related nonlinear partial differential equations

TL;DR

This paper introduces a family of generalized Donaldson's functionals for holomorphic vector bundles, leading to new complex vector bundle equations and exploring their solutions' uniqueness.

Contribution

It extends Donaldson's functionals to a broader class and formulates associated nonlinear PDEs, analyzing their solution uniqueness.

Findings

Defined a new family of Donaldson's functionals.

Derived vector bundle versions of complex k-Hessian equations.

Proved uniqueness of solutions to these equations.

Abstract

In this paper, we introduce a family of generalized Donaldson's functional on holomorphic vector bundles, whose Euler-Lagrange equations are a vector bundle version of the complex $k$-Hessian equations. We also discuss the uniqueness of solutions to these equations.