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Complexifications of nonnegatively curved manifolds

Complexifications of nonnegatively curved manifolds J. Eur. Math. Soc. 5, 69–94 (2003) Digital Object Identifier (DOI) 10.1007/s10097-002-0044-y Burt Totaro Complexifications of nonnegatively curved manifolds Received December 15, 2000 / final version received April 17, 2002 Published online August 15, 2002 –  Springer-Verlag & EMS 2002 Define a good complexification of aclosedsmoothmanifold 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 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of the European Mathematical Society Springer Journals

Complexifications of nonnegatively curved manifolds

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Publisher
Springer Journals
Copyright
Copyright © 2003 by Springer-Verlag Berlin Heidelberg & EMS
Subject
Mathematics; Mathematics, general
ISSN
1435-9855
DOI
10.1007/s10097-002-0044-y
Publisher site
See Article on Publisher Site

Abstract

J. Eur. Math. Soc. 5, 69–94 (2003) Digital Object Identifier (DOI) 10.1007/s10097-002-0044-y Burt Totaro Complexifications of nonnegatively curved manifolds Received December 15, 2000 / final version received April 17, 2002 Published online August 15, 2002 –  Springer-Verlag & EMS 2002 Define a good complexification of aclosedsmoothmanifold 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

Journal

Journal of the European Mathematical SocietySpringer Journals

Published: Mar 1, 2003

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