Complexifications of nonnegatively curved manifolds

Abstract

Define a good complexification of a closed smooth manifold M to be a smooth affine algebraic variety U over the real numbers such that M is diffeomorphic to U(R) and the inclusion U(R) → U(C) is a homotopy equivalence. Kulkarni showed that every manifold which has a good complexification has nonnegative Euler characteristic [16]. We strengthen his theorem 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 Pontrjagin numbers are zero (see Theorem 1.1 and, for a stronger statement, Theorem 2.1). We also construct a new class of manifolds with good complexifications. As a result, all known closed manifolds which have Riemannian metrics of nonnegative sectional curvature, including those found by Cheeger [5] and Grove and Ziller [11], have good complexifications. We can in fact ask whether a closed manifold has a good complexification if and only if it has a Riemannian metric of nonnegative sectional curvature. The question is suggested by the work of Lempert and Szőke [17]. Lempert and Szőke, and independently Guillemin and Stenzel [12], constructed a canonical complex analytic structure on an open subset of the tangent bundle of M, given a real analytic Riemannian metric on M. We say that a real analytic Riemannian manifold has entire Grauert tube if this complex structure is defined on the whole tangent bundle TM. Lempert and Szőke found that every Riemannian manifold with entire Grauert tube has nonnegative sectional curvature. Also, a conjecture by Burns [4] predicts that for every closed Riemannian manifold M with entire Grauert tube, the complex manifold TM is an affine algebraic variety in a natural way; if this is correct, the complex manifold TM would be a good complexification of M in the above sense. The problem remains open at this writing despite the paper of Aguilar and Burns [2], because the proof of Proposition 2.1 there is not yet generally accepted. We can also ask how many Riemannian manifolds have entire Grauert tube. It is known to be a strong restriction: Szőke showed that of all surfaces of revolution diffeomorphic to the 2-sphere, only a one-parameter family of metrics of given volume have entire Grauert tube [22]. Aguilar showed that the quotient of a Riemannian manifold with entire Grauert tube by a group of isometries acting freely also has entire Grauert tube [1]. All known manifolds with entire Grauert