Complex nilmanifolds with constant holomorphic sectional curvature

TL;DR

This paper confirms a longstanding conjecture in complex geometry for complex nilmanifolds, showing that such manifolds with constant holomorphic sectional curvature are either Kähler or Chern flat, depending on the curvature value.

Contribution

It proves the conjecture for complex nilmanifolds, a specific class of complex manifolds, extending the understanding of the curvature properties in higher dimensions.

Findings

Nilmanifolds with non-zero constant holomorphic sectional curvature are Kähler.

Nilmanifolds with zero constant holomorphic sectional curvature are Chern flat.

The conjecture holds true within the class of complex nilmanifolds.

Abstract

A well known conjecture in complex geometry states that a compact Hermitian manifold with constant holomorphic sectional curvature must be K\"ahler if the constant is non-zero and must be Chern flat if the constant is zero. The conjecture is confirmed 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 the class of complex nilmanifolds and confirm the conjecture in that case.