Invariant polynomials on compact complex manifolds
In Invent. Math. 73, 437-443 (1983; Zbl 0506.53030)] the first author introduced a new obstruction for a compact complex manifold M (dim M\(=n)\) with positive \(c_ 1(M)\) to admit a Kähler-Einstein metric. The obstruction is a character \(f: h(M)\to {\mathbb{C}},\) \(h(M)\) being the complex Lie algebra of all holomorphic vector fields of M. In the paper under review the authors interpretate f in terms of secondary characteristic classes of Chern-Simons and Cheeger-Simons. Moreover let H(M) be the complex Lie group of all automorphisms of M and let \(I^ p(G)\) denote the set of all holomorphic G-invariant symmetric polynomials of degree p, for any complex Lie group G; the authors introduce and study a linear map \(F: I^{n+k}(GL(n,{\mathbb{C}}))\to I^ k(H(M))\) which generalizes f, since \(f=F(c_ 1^{n+1})\), up to a constant.
Lie algebra of holomorphic vector fields
scientific article; zbMATH DE number 3941946
scientific article