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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.

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    $\begingroup$ A torus that is almost like a sphere, then take the product with a sphere $\endgroup$
    – Martin de Borbon
    Jan 7, 2019 at 23:31
  • $\begingroup$ Martin, I actually mean K is neither non-negative nor non-positive. Your example is a product Kaehler manifold. K is essentially the pullback by the projection (onto the sphere) of the canonical line bundle of the 2-sphere. So K is non-positive. I guess product manifold does not offer a good example for my question. I am sorry that my question is not very clear. $\endgroup$
    – Wai
    Jan 8, 2019 at 13:49
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    $\begingroup$ You can replace the torus with a genus 2 Riemann surface, a sphere with two very small handles attached $\endgroup$
    – Martin de Borbon
    Jan 8, 2019 at 16:10
  • $\begingroup$ Martin, that can be an example. But as you shrink the volume of region having negative eigenvalues of the Ricci tensor, the absolute value of the negative eigenvalues becomes large. That's not what I want, as I have explained in the note added for the first condition. Also, the first Betti number of the Kaehler manifold (the fourth condition added later) is not zero. In fact, I guess the fourth condition will be a consequence of the first three. (Think of Bochner and Kodaira vanishing theorems, though they aren't valid here.) I think product manifold is not good. Other examples? $\endgroup$
    – Wai
    Jan 8, 2019 at 18:46
  • $\begingroup$ By "non-negative" (line bundle), do you mean nef? $\endgroup$
    – macbeth
    Jan 17, 2019 at 20:35

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