Complexifications of nonnegatively curved manifolds
A good complexification of a closed smooth manifold \(M\) is a smooth affine algebraic variety \(U\) over the real numbers such that \(M\) is diffeomorphic to \(U(\mathbb R)\) and the inclusion \(U(\mathbb R)\to U(\mathbb C)\) is a homotopy equivalence. \textit{R. S. Kulkarni} showed in [Invent. Math. 44, 49--54 (1978; Zbl 0375.32007)] that every manifold which has a good complexification has nonnegative Euler characteristic. In this paper, the author strengthens the result to say that if the Euler characteristic is positive, then all the odd Betti numbers are zero; also, if the Euler characteristic is zero, then all the Pontryagin numbers are zero. Another main result of this paper is that all of Cheeger's manifolds and all closed manifolds of cohomogeneity one have good complexifications. In fact, one gives good complexifications for a certain class of manifolds which includes both of these types: roughly, union of two disc bundles over manifolds with good complexifications. Thus, all known closed manifolds with matrics of nonnegative curvature have good complexifications. Next, one shows that \(S^1\) has only one good complexification up to isomorphism, and moreover, one shows that \(S^2\) an \(\mathbb R\mathbb P^2\) have only one good complexification if one strengthenes the assumption that \(U(\mathbb R)\to U(\mathbb C)\) is a homotopy equivalence to say \(U(\mathbb C)\) is diffeomorphic to the total space of the tangent bundle of \(M= U(\mathbb R)\). Finally, one proves that the automorphism group of the real affine variety \(S^2\) is the orthogonal group \(O(3)\) but also, one shows that the automorphism group of a good complexification can be noncompact or even infinite-dimensional.
scientific article; zbMATH DE number 1968516
scientific article