My question is about whether there are some known conditions on the sign of the Euler-Poincaré characteristic for Einstein manifolds in even dimensions.
In dimension $4$ some conditions on the sign of the Euler--Poincaré characteristic are easy consequences of the generalised Gauss-Bonnet theorem as shown in Bishop, R. L.; Goldberg, S. I., Some implications of the generalized Gauss-Bonnet theorem:
Theorem 1.1. A compact and oriented Riemannian manifold of dimension 4 whose sectional curvatures are non-negative or nonpositive has non-negative Euler-Poincaré characteristic. If the sectional curvatures are always positive or always negative, the Euler-Poincaré characteristic is positive.
Theorem 1.2. In order that a 4-dimensional compact and orientable manifold M carry an Einstein metric, i.e., a Riemannian metric of constant Ricci or mean curvature R, it is necessary that its Euler-Poincaré characteristic be non-negative.
Then in dimension $6$ in Gray, Alfred, The six dimensional Gauss Bonnet integrand the author proves
Theorem 2. Let $M$ be a compact $6$-dimensional Einstein Kähler manifold. Assume that for each me $M$ the holomorphic sectional curvature assumes critical values on a triple of mutually orthogonal holomorphic sections in $M_m$. If $M$ has nonnegative (nonpositive) sectional curvatures then $\chi(M) \geq 0$ $(\chi(M) \leq 0)$
and states that the problem (of proving whether nonnegative or nonpositive sectional curvature implies $\chi(M) \geq 0$ or $\chi(M) \leq 0$, respectively) “[...] is not too far from being settled for compact 6-dimensional Einstein Kähler manifolds”.
The paper by A. Gray is from 1973 and I wasn't able to find any update on the matter.
Are there any results when one only assumes the manifold to be Kähler-Einstein in addition to having nonnegative sectional curvatures in dimension $6$? What if one removes the hypothesis of nonnegative sectional curvatures? Has anything been shown in higher dimensions?
