The Demailly system for a direct sum of ample line bundles on Riemann surfaces

TL;DR

This paper proves the existence of smooth solutions for a system of equations related to ample line bundles on Riemann surfaces, advancing the understanding of Griffiths' conjecture and reducing the problem to an a priori estimate.

Contribution

It establishes the existence of solutions for Demailly's system on Riemann surfaces and links the problem for vector bundles to an a priori estimate via Leray-Schauder degree theory.

Findings

Proved smooth solutions exist for the system on Riemann surfaces.

Reduced the general vector bundle problem to an a priori estimate.

Connected the problem to Leray-Schauder degree theory.

Abstract

We prove that a system of equations introduced by Demailly (to attack a conjecture of Griffiths) has a smooth solution for a direct sum of ample line bundles on a Riemann surface. We also reduce the problem for general vector bundles to an a priori estimate using Leray-Schauder degree theory.