Although compact Kähler manifolds of constant holomorphic sectional curvature have been classified [S. Kobayashi and K. Nomizu, Foundations of differential geometry, vol. II (1969; Zbl 0175.485)], little is known of the more general Hermitian case. The present author shows that there are examples of compact non-Kähler Hermitian manifolds of constant zero holomorphic sectional curvature in every dimension above 2. Exactly he proves: Let G be a complex Lie group and \(\Gamma\) \(\subset G\) a discrete subgroup. Then there is a G-invariant Hermitian metric on \(M=G/\Gamma\) with vanishing curvature. Moreover, it is Kähler if and only if G is Abelian.
The author’s main result is the following theorem: Let M be a compact Hermitian manifold of constant holomorphic sectional curvature \(=k\). Let \(P_ m\) be the mth plurigenus and \(Q_ m\) be the mth dual plurigenus of M. Then a) \(k>0\Rightarrow P_ m=0\), \(\forall m>0\); b) \(k=0\Rightarrow either\) \(P_ m=0\), \(\forall m>0\), or \(P_ m=Q_ m\), \(\forall m>0\), and \(P_ m\in \{0,1\}\).