Compact Hermitian surfaces with pointwise constant Gauduchon holomorphic sectional curvature

TL;DR

This paper classifies compact Hermitian surfaces with pointwise constant Gauduchon holomorphic sectional curvature, showing they are either Kähler or specific Hopf surfaces, and extends the classification to more general connections.

Contribution

It provides a complete classification of such surfaces under Gauduchon and canonical connections, extending previous results in the field.

Findings

Hermitian surfaces with constant holomorphic sectional curvature are either Kähler or specific Hopf surfaces.

The classification extends to two-parameter canonical connections.

A special case with Lichnerowicz curvature implies the surface is Kähler.

Abstract

Motivated by a recent work of Chen-Zheng [8] on Strominger space forms, we prove that a compact Hermitian surface with pointwise constant holomorphic sectional curvature with respect to a Gauduchon connection $\nabla^t $ is either K\"ahler, or an isosceles Hopf surface with an admissible metric and $t=-1$ or $t=3$. In particular, a compact Hermitian surface with pointwise constant Lichnerowicz holomorphic sectional curvature is K\"ahler. We further generalize the result to the case for the two-parameter canonical connections introduced by Zhao-Zheng [30], which extends a previous result by Apostolov-Davidov-Mu\v{s}karov [2].