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arXiv · 2501.00810

On solvmanifolds with complex commutator and constant holomorphic sectional curvature

Xin Huang·Fangyang Zheng
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Abstract

An old open question in non-Kähler geometry predicts that any compact Hermitian manifold with constant holomorphic sectional curvature must be Kähler 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.

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Xin Huang, Fangyang Zheng. 2025-01-01. On solvmanifolds with complex commutator and constant holomorphic sectional curvature. https://arxiv.org/abs/2501.00810

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