# examples of Kähler manifolds with trivial odd Betti numbers and first Chern classes

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!

• $K3$ surfaces ? – Alex Degtyarev May 24 '14 at 9:42
• There are infinitely many examples: for instance, complete intersections $X\subset\mathbb{P}^n$ of type $(d_1,\ldots,d_c)$, with $\dim X=n-c$ even and $d_1+\ldots +d_c=n+1$ (so that $X$ is C-Y). – abx May 24 '14 at 10:20
• It is not possible to add accents and related using TeX-style syntax. (For the body html-entities would work.) – user9072 May 24 '14 at 10:29
• Rigid Calabi--Yau threefolds (e.g. arxiv.org/abs/1102.1854) are odd dimensional examples. – Sasha May 24 '14 at 10:30
• You should look at the article of Beauville, "Variétés Kähleriennes dont la première classe de Chern est nulle", J. Differential Geom. Volume 18, Number 4 (1983), 755-782. – Jason Starr May 24 '14 at 10:33

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, $$c_1(X) = c_1(T_{X}) = -c_1(\Omega_{X}) = c_1(\bigwedge^2\Omega_{X})= c_1(\omega_{X}) = c_1(\mathcal{O}_X) = 0.$$
• Thanks for your comments. My primary question is that, for any $n$, whether or not there exists such an complex $n$-dimensional example. – Kevin May 24 '14 at 12:27
• For that, as abx wrote, just take a smooth Calabi-Yau complete intersection of even dimension. For instance if you take a degree $4$ smooth surface in $\mathbb{P}^3$ (which is a $K3$ surface) you get an instance of my example. – F_L May 24 '14 at 14:17