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Positive vector bundles and harmonic maps. (English) Zbl 0674.58015

Concerning the holomorphicity of a mapping between complex manifolds, the authors have gained the following results:
(1) If f: (N,g)\(\to (M,h)\) is a harmonic map between Kähler manifolds where N is compact of dimension n and (a) Ric \(N\geq 0\) and \(=0\) somewhere, (b) \(E:=f^*(F)\) \((F:=TM_{(1,0)}\), T is the tangent functor) n-semipositive, then f is holomorphic.
(2) Result (1) is formulated and proved in the case of boundary conditions.
(3) If (M,h) is a Kähler manifold with positive bisectional curvature, (N,g) is another complete Kähler manifold and f: (N,g)\(\to (M,h)\) is a stable harmonic map, then under some further delicate analytic conditions f is holomorphic again.
(4) Result (3) is applied to the special case \(N={\mathbb{C}}\) for drawing further conclusions. It is also shown that for pluriharmonic f one of the crucial assumptions of (3) is always satisfied.
Reviewer: J.Szilasi

MSC:

58E20 Harmonic maps, etc.
58C10 Holomorphic maps on manifolds
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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