# On the Structure of Hermitian Manifolds with Semipositive Griffiths Curvature

## TL;DR

This paper investigates the geometric structure of compact Hermitian manifolds with semipositive Griffiths curvature, revealing how small metric deformations lead to holomorphic distributions and group actions, advancing understanding of their complex geometry.

## Contribution

It establishes that small metric deformations produce holomorphic, integrable distributions from null spaces of the Chern-Ricci form, linking curvature conditions to geometric and group action structures.

## Key findings

- Null spaces generate holomorphic, integrable distributions after deformation
- Universal cover admits a complex Lie group action
- Uses Hermitian Curvature Flow and torsion-twisted connection analysis

## Abstract

In this paper we establish partial structure results on the geometry of compact Hermitian manifolds of semipositive Griffiths curvature. We show that after appropriate arbitrary small deformation of the initial metric, the null spaces of the Chern-Ricci two-form generate a holomorphic, integrable distribution. This distribution induces an isometric, holomorphic, almost free action of a complex Lie group on the universal cover of the manifold. Our proof combines the strong maximum principle for the Hermitian Curvature Flow (HCF), new results on the interplay of the HCF and the torsion-twisted connection, and observations on the geometry of the torsion-twisted connection on a general Hermitian manifold.