On solvmanifolds with complex commutator and constant holomorphic sectional curvature

TL;DR

This paper proves that all solvmanifolds with complex commutator and constant holomorphic sectional curvature are either K"ahler or Chern flat, advancing understanding of Hermitian space forms in non-K"ahler geometry.

Contribution

It confirms the conjecture for a broad class of solvmanifolds, extending previous results from nilmanifolds to solvmanifolds with complex commutator.

Findings

Solvmanifolds with complex commutator and constant holomorphic sectional curvature are K"ahler or Chern flat.

Extends earlier results from nilmanifolds to solvmanifolds.

Supports the conjecture in higher dimensions for this class.

Abstract

An old open question in non-K\"ahler geometry predicts that any compact Hermitian manifold with constant holomorphic sectional curvature must be K\"ahler or Chern flat. The conjecture is known to be true in dimension $2$ due to the work by Balas-Gauduchon and Apostolov-Davidov-Muskarov in the 1980s and 1990s, but is still open in dimensions $3$ or higher, except in several special cases. The difficulty in this quest for `Hermitian space forms' is largely due to the algebraic complicity or lack of symmetry for the curvature tensor of a general Hermitian metric. In this article, we confirm the conjecture for all solvmanifolds with complex commutator, extending earlier result on nilmanifolds by Li and the second named author.