An analytic proof of Griffiths' conjecture on compact Riemann surfaces

Abstract

Griffiths' conjecture asserts that a holomorphic vector bundle is ample if and only if it admits a Hermitian metric with positive curvature. In this paper, we present a new proof of this conjecture on compact Riemann surfaces using a system of PDEs introduced by Demailly. Our argument combines techniques developed by Uhlenbeck-Yau for Hermitian-Einstein metrics with Pingali's reduction of the problem to an a priori estimate.

arXiv commentAccepted version. Minor revisions: corrected an issue in the argument, clarified the statement of the main theorem, fixed typos, and added a brief remarks section at the end of the paper. 12 pages