Kähler parabolicity and the Euler number of compact manifolds of non-positive sectional curvature

Abstract

Let M 2n be a compact Riemannian manifold of non-positive sectional curvature. It is shown that if M 2n is homeomorphic to a Kähler manifold, then its Euler number satisfies the inequality (−1) n χ(M 2n ) ≥ 0. M 2n of negative sectional curvature must satisfy the inequality (−1) n χ(M 2n ) > 0. This conjecture is true in dimensions 2 and 4 [Ch] and it has been verified in the Kähler case for all n by Gromov [G] and Stern [S] (the work in [S] also uses results of Greene and Wu; see [GW], p.183-215). Gromov's arguments are rather general and establish the following result: "Let M 2n be a compact Riemannian manifold of negative curvature. If M 2n is homotopy equivalent to a compact Kähler manifold then (−1) n χ(M 2n ) > 0"; see [G], Theorem 0.4.A and Example (a), p.265.A companion conjecture asserts that, if the sectional curvature of a Riemannian manifold M 2n is assumed to be only non-positive, then the Euler number must satisfy (−1) n χ(M 2n ) ≥ 0. Again, this second conjecture is known to be true in dimensions two and four [Ch]. The aim of this paper is to establish its validity for all n in the Kähler case, thus complementing the above result of Gromov:Main Theorem. Let M 2n be a compact Riemannian manifold of non-positive curvature. If M 2n is homeomorphic to a Kähler manifold, then the Euler number of M 2n satisfies the inequality (−1) n χ(M 2n ) ≥ 0.Remark. The proof of the Main Theorem works if the two manifolds are assumed to be only homotopy equivalent but, in view of Farrell and Jones [FJ], the man-

Von Neumann dimension, Hodge index theorem and geometric applications

“…This is a particular case of an old problem that goes back to Hopf, see [10] and the references cited there. Indeed Hopf conjectured that for a compact, [5] and Jost-Zuo in [9] showed that (−1) m χ(M ) ≥ 0 when sec g (M ) ≤ 0. We recall now some of the main definitions and results from [5], [8] and [9].…”

“…Indeed Hopf conjectured that for a compact, [5] and Jost-Zuo in [9] showed that (−1) m χ(M ) ≥ 0 when sec g (M ) ≤ 0. We recall now some of the main definitions and results from [5], [8] and [9]. In [8] Gromov proved the following: In [5] and [9] the authors introduced the next definition, which includes as a particular case, the above definition of Gromov: In this setting the authors proved in [5] and [9] the following: In what follows we will show that these questions are deeply connected with the non vanishing of the signature of M .…”

“…k (M ). As shown in [5] and [9] we have b Let (M, h) be a compact, Kähler, Kähler-parabolic manifold of complex dimension 2m. Assume that σ(M ) = 0.…”

Infinite Dimensionality of the Middle $L^2$-cohomology on Non-compact Kähler Hyperbolic Manifolds

“…[9], [13]) play an essential role, even in the special case of the ball one has to use some deep theorems such as Atiyah's L 2 −index theorem [1] and the Hirzebruch proportionality principle [11]. Vanishing theorems in [8], [10] have been extended to certain "non-elliptic" cases in [3], [12], [15]. A typical example of those results is C n equipped with the Euclidean metric.…”

The first L2-Betti number of certain homogeneous spaces, including classifying spaces for variations of Hodge structures

“…Besides offering the possibility to extend Hodge theory to the noncompact case, it can be used to obtain topological information about compact quotients of X by the L 2 -index theorem of Atiyah [1]. Based on this theorem Dodziuk-Singer [4] [12] suggested to use L 2 -cohomology to approach the Hopf conjecture concerning the Euler characteristic of Riemannian manifolds of non-positive curvature, which has been verified in several special but important cases [2], [3], [5], [8], [9] and [10]. For a detailed discussion, we refer to the comprehensive volume on L 2 -cohomology by W. Lück [11].…”