The Demailly systems with the Vortex ansatz

TL;DR

This paper proves the existence of smooth solutions to certain Hermitian-Yang-Mills type equations for vortex bundles over projective manifolds, advancing understanding of Griffiths positivity and Hartshorne ampleness.

Contribution

It demonstrates the solvability of two proposed Demailly systems for vortex bundles using the continuity method, providing new insights into complex differential geometry.

Findings

Smooth solutions exist for the studied systems on vortex bundles.

The continuity method effectively proves existence of solutions.

Advances the understanding of Griffiths positivity in vector bundles.

Abstract

For an arbitrary-rank vector bundle over a projective manifold, J.-P. Demailly proposed several systems of equations of Hermitian-Yang-Mills type for the curvature tensor to settle a conjecture of Griffiths on the equivalence of Hartshorne ampleness and Griffiths positivity. In this article, we have studied two proposed systems and proved that these equations have smooth solutions for the Vortex bundle using the continuity method.