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Secondary characteristic classes of a Kählerian manifold with vanishing real first Chern class. (English) Zbl 0702.53041

The main purpose of this article is to construct for a Kähler manifold (M,g) with vanishing real first Chern class \((c_ 1(M)_ R=0)\), a cohomology class h(M,g) in \(H^*(M,R/Z)\). The resulting objects may be closely related to secondary characteristic classes in the sense of D. Lehmann [Ann. Inst. Fourier 24, No.3, 267-306 (1974; Zbl 0268.57009)].
The main theorem is the following Theorem A. Suppose \(c_ 1(M)=0\). Then the following conditions are equivalent. (1) (M,g) is Ricci-flat and \(h(M,g)=0\). (2) The homology group of (M,g) is contained in SU(m). (3) The canonical bundle \(K_ M\) of M has a non-zero parallel holomorphic section. Especially, for a compact Kählerian manifold the author proves that \(h(M,g_ 0)\in H^ 1(M,R/Z)\) does not depend on the choice of \(g_ 0\).
Reviewer: N.Bokan

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
57R20 Characteristic classes and numbers in differential topology
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