On mixed curvature for Hermitian manifolds

TL;DR

This work studies mixed curvature $\mathcal{C}_{\alpha,\beta}$ on Hermitian manifolds, combining the first Chern Ricci curvature with holomorphic sectional curvature. It first proves that compact Hermitian surfaces with pointwise constant $\mathcal{C}_{\alpha,\beta}$ are Kähler except in the special case $c=0$ with $2\alpha+\beta=0$, where isosceles Hopf surfaces may occur, and then extends to higher dimensions under locally conformal Kähler or Kähler-like assumptions with analogous conclusions. It also shows that for locally conformal Kähler manifolds with constant $\mathcal{C}_{\alpha,\beta}$, certain sign conditions force Kähler metrics; in the CK-like setting, similar results hold without compactness. Finally, the paper proves a Kodaira-dimension vanishing result: if $\mathcal{C}_{\alpha,\beta}$ is semi-positive but not identically zero with $\beta\ge0$ and $\alpha(n+1)+2\beta>0$, then $\kappa(M)=-\infty$, linking mixed-curvature positivity to global algebro-geometric structure.

Abstract

In this paper, we consider {\em mixed curvature} $\mathcal{C}_{α,β}$ for Hermitian manifolds, which is a convex combination of the first Chern Ricci curvature and holomorphic sectional curvature introduced by Chu-Lee-Tam \cite{CLT}. We prove that if a compact Hermitian surface with constant mixed curvature $c$, then the Hermitian metric must be Kähler unless $c=0$ and $2α+β=0$, which extends a previous result by Apostolov-Davidov-Muškarov. For the higher-dimensional case, we also partially classify compact locally conformal Kähler manifolds with constant mixed curvature. Lastly, we prove that if $β\geq0, α(n+1)+2β>0$, then a compact Hermitian manifold with semi-positive but not identically zero mixed curvature has Kodaira dimension $-\infty$.

Theorems & Definitions (14)

• Conjecture 1.1
• Theorem 1.1: Apostolov-Davidov-Muškarov
• Conjecture 1.2
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Lemma 2.1: ADM
• Proposition 3.1