Canonical metric connections with constant holomorphic sectional curvature

TL;DR

This work addresses when canonical metric connections $D^r_s$ on compact Hermitian manifolds have constant holomorphic sectional curvature $c$. It leverages Gauduchon’s two-parameter family and the Chen–Nie curve $\Gamma$ to distinguish zero-curvature from nonzero-curvature cases, deriving strong rigidity results for two special classes: complex nilmanifolds with nilpotent $J$ and non-balanced Bismut torsion-parallel manifolds. The authors combine curvature decompositions linking $R^D$, $R^c$, and $R^b$ with torsion identities to show that, in the nilmanifold case, either the metric is flat (non-Chern case) or Chern flat (Chern case); in the non-balanced BTP case, $c=0$ with $(r,s)\in\Gamma$, while in balanced BTP threefolds the geometry is either Kähler or Chern flat with a precise $(r,s)$ and manifold structure. Additionally, they analyze standard Hopf manifolds to illustrate the role of the Chen–Nie curve in zero-curvature scenarios and to confirm that, outside $\Gamma$, constant curvature is not possible; within $\Gamma$, vanishing curvature occurs but flatness is dimension-dependent. Overall, the paper advances a targeted verification of Conjecture 5 in meaningful non-Kähler contexts and clarifies the geometric consequences of constant holomorphic sectional curvature for Gauduchon connections.

Abstract

We consider the conjecture of Chen and Nie concerning the space forms for canonical metric connections of compact Hermitian manifolds. We verify the conjecture for two special types of Hermitian manifolds: complex nilmanifolds with nilpotent $J$, and non-balanced Bismut torsion-parallel manifolds.

Theorems & Definitions (6)

• Conjecture 1: Constant Holomorphic Sectional Curvature Conjecture
• Conjecture 2
• proof : Proof of Theorem 7 for nilmanifolds.
• proof : Proof of Theorem 7 for non-balanced BTP manifolds.
• proof
• proof : Proof of Theorem 7 for balanced BTP threefolds.