An analytic proof of Griffiths' conjecture on compact Riemann surfaces
Differential Geometry
2025-12-25 v2
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.
Cite
@article{arxiv.2509.23201,
title = {An analytic proof of Griffiths' conjecture on compact Riemann surfaces},
author = {Rei Murakami},
journal= {arXiv preprint arXiv:2509.23201},
year = {2025}
}
Comments
Accepted 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