Example of a Kähler manifold with certain properties

Question

I am looking for compact Kähler manifolds of dimension $3$ with the following 2 properties:

1. $c_1(K_X)=c[\omega],c>0$ where $\omega$ is the Kähler form on $X$.

2. $1+h^{0,3}+h^{1,1}=h^{0,1}$

It's easy to find example satisfying the first condition. One can take $X=X_1\times X_2\times X_3$ where $X_i$s are compact Riemann surfaces of same genus $g\geq 2$. Another set of examples would be hypersurfaces in $\mathbb{CP}^4$ of very high degree, i.e., holomorphic sections of $\mathcal{O}(d)\rightarrow \mathbb{CP}^4$, we have $K_X=\mathcal{O}(d-5)|_X$. The second condition seems harder to check. Any help is appreciated.

Answer 1

What about taking a product $X=Y\times \Sigma$, where $Y$ is a fake projective plane and $\Sigma$ is a surface of genus $3$? We have $h^{1,1}(X)=2$, $h^{0,1}(X)=3$, and $h^{0,3}(X)=0$ by Kuneth formula. Condition 1) holds since this is a $3$-fold of general type with ample canonical bundle.