A note on Demailly's approach towards a conjecture of Griffiths

The Demailly systems with the Vortex ansatz

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.

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

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.

Demailly's notion of algebraic hyperbolicity: geometricity, boundedness, moduli of maps

Demailly's conjecture, which is a consequence of the Green-Griffiths-Lang conjecture on varieties of general type, states that an algebraically hyperbolic complex projective variety is Kobayashi hyperbolic. Our aim is to provide evidence for Demailly's conjecture by verifying several predictions it makes. We first define what an algebraically hyperbolic projective variety is, extending Demailly's definition to (not necessarily smooth) projective varieties over an arbitrary algebraically closed field of characteristic zero, and we prove that this property is stable under extensions of algebraic…

希格斯-德麦利系统与希格斯丛的正性

Higgs-Demailly System and Positivity of Higgs Bundles

该研究证明紧黎曼曲面上Higgs-Demailly系统终端参数处存在光滑容许解等价于希格斯丛H-丰富，通过Griffiths正的Hitchin-Simpson曲率刻画H-丰富性，扩展了相关解析方法并解决了标量下界问题。

关于紧黎曼面上Griffiths猜想的解析证明

An analytic proof of Griffiths' conjecture on compact Riemann surfaces

本文通过 Demailly 引入的 PDE 系统，结合 Uhlenbeck-Yau 和 Pingali 的技术，给出了紧黎曼面上 Griffiths 猜想的解析证明。

Logarithmic Jet Bundles and Applications

We generalize Demailly's construction of projective jet bundles and strictly negatively curved pseudometrics on them to the logarithmic case. We establish this logarithmic generalization explicitly via coordinates, just as Noguchi's generalization of the jets used by Green-Griffiths. As a first application, we give a metric proof for the logarithmic version of Lang's conjecture concerning the hyperbolicity of complements of divisors in a semi-abelian variety as well as for the corresponding big Picard theorem.