Higa, Tatsuo Secondary characteristic classes of a Kählerian manifold with vanishing real first Chern class. (English) Zbl 0702.53041 Comment. Math. Univ. St. Pauli 39, No. 1, 69-80 (1990). 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 Keywords:Kähler manifold; Chern class; secondary characteristic classes Citations:Zbl 0291.57011; Zbl 0268.57009 PDFBibTeX XMLCite \textit{T. Higa}, Comment. Math. Univ. St. Pauli 39, No. 1, 69--80 (1990; Zbl 0702.53041)