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

The results of this paper are related to a well-known problem, attributed to Hopf, to the effect that the Euler number χ(M2n) of a compact Riemannian manifold M2n of negative sectional curvature must satisfy the inequality (−1)nχ(M2n) > 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 M2n be a compact Riemannian manifold of negative curvature. If M2n is homotopy equivalent to a compact Kähler manifold then (−1)nχ(M2n) > 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 M2n is assumed to be only non-positive, then the Euler number must satisfy (−1)nχ(M2n) ≥ 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 M2n be a compact Riemannian manifold of non-positive curvature. IfM2n is homeomorphic to a Kähler manifold, then the Euler number ofM2n satisfies the inequality (−1)nχ(M2n) ≥ 0.