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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!

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    $\begingroup$ $K3$ surfaces ? $\endgroup$
    – Alex Degtyarev
    Commented May 24, 2014 at 9:42
  • $\begingroup$ 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). $\endgroup$
    – abx
    Commented May 24, 2014 at 10:20
  • $\begingroup$ It is not possible to add accents and related using TeX-style syntax. (For the body html-entities would work.) $\endgroup$
    – user9072
    Commented May 24, 2014 at 10:29
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    $\begingroup$ Rigid Calabi--Yau threefolds (e.g. arxiv.org/abs/1102.1854) are odd dimensional examples. $\endgroup$
    – Sasha
    Commented May 24, 2014 at 10:30
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    $\begingroup$ 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. $\endgroup$
    – Jason Starr
    Commented May 24, 2014 at 10:33

1 Answer 1

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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, $$c_1(X) = c_1(T_{X}) = -c_1(\Omega_{X}) = c_1(\bigwedge^2\Omega_{X})= c_1(\omega_{X}) = c_1(\mathcal{O}_X) = 0.$$

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  • $\begingroup$ Thanks for your comments. My primary question is that, for any $n$, whether or not there exists such an complex $n$-dimensional example. $\endgroup$
    – Kevin
    Commented May 24, 2014 at 12:27
  • $\begingroup$ 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. $\endgroup$
    – Puzzled
    Commented May 24, 2014 at 14:17

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