Compact Hermitian manifolds of constant holomorphic sectional curvature. (English) Zbl 0575.53044
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 \(P_ m\in \{0,1\}\).
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 \(P_ m\in \{0,1\}\).
Reviewer: St.Ianus
MSC:
53C55 | Global differential geometry of Hermitian and Kählerian manifolds |
Keywords:
non-Kähler Hermitian manifolds; holomorphic sectional curvature; complex Lie group; G-invariant Hermitian metricCitations:
Zbl 0175.485References:
[1] | Gauduchon, P.: Fibrés hermitiens à endomorphisme de Ricci non négatif. Bull. Soc. Math. Fr.105, 113-140 (1977) · Zbl 0382.53045 |
[2] | Gauduchon, P.: Le théorème de l’excentricité nulle. Comptes Rendus (Acad. Sc. Paris), t.285, Serie A (1977), 387-390 · Zbl 0362.53024 |
[3] | Goldberg, S.I.: Tensorfields and curvature in Hermitian manifolds with torsion. Ann. Math.63, 64-76 (1956) · Zbl 0070.17105 · doi:10.2307/1969990 |
[4] | Goldberg, S.I.: Curvature and Homology. New York: Academic Press 1962 · Zbl 0105.15601 |
[5] | Kobayashi, S.: Hyperbolic Maniforlds and Holomorphic Mappings. New York: Dekker 1970 · Zbl 0207.37902 |
[6] | Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, Vol. II. New York: Wiley 1969 · Zbl 0175.48504 |
[7] | Kobayashi, S., Wu, H.: On holomorphic sections of certain Hermitian vector bundles. Math. Ann.189, 1-4 (1970) · Zbl 0189.52201 · doi:10.1007/BF01350196 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.