Pluriclosed manifolds with constant holomorphic sectional curvature

TL;DR

This paper investigates a longstanding conjecture in complex geometry, confirming it for pluriclosed manifolds that are Strominger K"ahler-like, thus advancing understanding in higher-dimensional cases.

Contribution

It proves the conjecture for Strominger K"ahler-like pluriclosed manifolds, a special class where the Strominger connection exhibits K"ahler symmetries.

Findings

Confirmed the conjecture for Strominger K"ahler-like manifolds.

Extended the validity of the conjecture to higher dimensions.

Provided new insights into the structure of pluriclosed manifolds.

Abstract

A long-standing conjecture in complex geometry says that a compact Hermitian manifold with constant holomorphic sectional curvature must be K\"ahler when the constant is non-zero and must be Chern flat when the constant is zero. The conjecture is known in complex dimension $2$ by the work of Balas-Gauduchon in 1985 (when the constant is zero or negative) and by Apostolov-Davidov-Muskarov in 1996 (when the constant is positive). For higher dimensions, the conjecture is still largely unknown. In this article, we restrict ourselves to pluriclosed manifolds, and confirm the conjecture for the special case of Strominger K\"ahler-like manifolds, namely, for Hermitian manifolds whose Strominger connection (also known as Bismut connection) obeys all the K\"ahler symmetries.