On mixed curvature for Hermitian manifolds

Kai Tang

Vol. 344 (2026), No. 1, 201–218

DOI: 10.2140/pjm.2026.344.201

Abstract

We consider mixed curvature ${\mathcal{𝒞}}_{\alpha ,\beta }$ for Hermitian manifolds, which is a convex combination of the first Chern Ricci curvature and holomorphic sectional curvature introduced by Chu, Lee and Tam (2022). We prove that if a compact Hermitian surface has constant mixed curvature $c$, then the Hermitian metric must be Kähler unless $c=0$ and $2\alpha +\beta =0$, which extends a previous result by Apostolov, Davidov and Mushkarov (1996). For the higher-dimensional case, we also partially classify compact locally conformal Kähler manifolds with constant mixed curvature. Lastly, we prove that if $\beta \ge 0$ and $\alpha (n+1)+2\beta >0$, then a compact Hermitian manifold with semipositive but not identically zero mixed curvature has Kodaira dimension $-\infty$.

Keywords

Hermitian manifold, holomorphic sectional curvature, Chern Ricci curvature

Mathematical Subject Classification

Primary: 32Q15, 53C55

Milestones

Received: 9 March 2025

Revised: 1 April 2026

Accepted: 25 May 2026

Published: 10 August 2026