Locally symmetric and rigid factors for complex manifolds via harmonic maps. (English) Zbl 0828.53058

Frankel, Sidney

Ann. Math. (2) 141, No. 2, 285-300 (1995).

The isometry group \(\text{Aut} (M)\) of a compact Riemannian manifold \(M\) is very well studied. If \(M\) is compact with negative Ricci curvature, S. Bochner and K. Yano proved that \(\text{Aut} (M)\) is a finite group [see S. Kobayashi and K. Nomizu, Foundations of differential geometry, Wiley Interscience, New York, Vol. 1 (1963; Zbl 0119.375), Vol. 2 (1969; Zbl 0175.485)]. On the other side the universal cover \(\widetilde {M}\) can have a very large isometry group. In the context of complex manifolds we have an analogous situation. If \(M\) is a compact complex manifold with negative first Chern class its group of holomorphic automorphisms is finite.
In the abstract the author says: “We obtain a complete description of the Lie algebra of complete holomorphic vector fields on the universal cover of a compact complex manifold with negative first Chern class. The main tools are an equivariance result we prove for harmonic maps and the rigidity theory for harmonic maps from Kähler manifolds to locally symmetric spaces.”
One of the main theorems in the paper is that for a compact complex manifold with \(c_1(M) < 0\) there is a holomorphic splitting of the finite cover of \(M\), \(M' = M_1 \times M_2\), such that \(M_1\) is locally symmetric and \(M_2\) is locally rigid.

Reviewer: N.Blažić (Beograd)

MSC:

53C55	Global differential geometry of Hermitian and Kählerian manifolds
58E20	Harmonic maps, etc.
32Q99	Complex manifolds

Keywords:

holomorphic vector fields; compact complex manifold; Chern class

Citations:

Zbl 0119.375; Zbl 0175.485