TITLE: examples of Kähler manifolds with trivial odd Betti numbers and first Chern classes
QUESTION [5 upvotes]: To my limited knowledge, many compact Kähler manifolds have trivial odd Betti numbers. For instance, flag manifolds $G/P$，where $G$ is a semisimple complex Lie group and $P$ a parabolic subgroup, and Fano contact manifolds. But all these manifolds are Fano, i.e., with positive first Chern class. 
My question is, does there exist a compact Kähler manifold with trivial odd Betti numbers so that its first Chern class is also trivial?
Thanks in advance!

REPLY [7 votes]: Let $X$ be a $K3$ surface. That is $\omega_{X} = \bigwedge^2\Omega_{X}\cong\mathcal{O}_X$ and $H^{1}(X,\mathcal{O}_X) = 0$. All $K3$ surfaces are simply connected. The Hodge diamond is completely determined:
$$
\begin{array}{ccccc}
 &  & 1 &  & \\
 & 0 &  & 0 & \\
1 &  & 20 &  & 1\\
 & 0 &  & 0 & \\
 &  & 1 &  & \\
\end{array}
$$
From this one can see that the Betti numbers are
$$b_0(X)=1,\: b_1(X)=0,\: b_2(X)= 22,\: b_3(X)=0,\: b_4(X)=1.$$ 
Over the complex numbers any $K3$ surface is Kähler. Finally, 
$$c_1(X) = c_1(T_{X}) = -c_1(\Omega_{X}) = c_1(\bigwedge^2\Omega_{X})= c_1(\omega_{X}) = c_1(\mathcal{O}_X) = 0.$$