A Kaehler manifold is a complex manifold which has a Kaehler metric and Ricci curvature tensor $R_{ij}$. The Ricci curvature tensor is a Hermitian matrix having real eigenvalues. My question is: Is there a compact Kaehler manifold $M$ of complex dimension $n$ (with $n>1$) such that the following conditions are satisfied:

(1) At most points of some region $U$ of $M$, the Ricci curvature tensor is neither positive nor negative. At these points, the Ricci tensor has positive and negative eigenvalues. Also, the Kaehler or Riemannian volume of this region is small. (Note added: Also, in certain sense, the negative eigenvalues should not be too large in absolute values.)

(2) Outside the region mentioned in (1), the Ricci tensor is positive (definite) and the Riemannian volume of this region (the region outside $U$) is comparatively large.

(3) The canonical line bundle $K$ of $M$ is neither positive nor negative. (Note that the curvature of $K$ is given by the Ricci curvature). (Note added: I actually mean,"The canonical line bundle is neither non-negative nor non-positive.")

Also, I would like to add a fourth condition, but it is optional and not necessary:

(4) The first Betti number $\beta_1(M)$ is zero.

In complex dimension $n=1$, the canonical line bundle of a compact Riemann surface is either positive, negative or trivial. So, I am asking examples of complex dimension greater than 1.