3
$\begingroup$

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!

$\endgroup$
  • 5
    $\begingroup$ $K3$ surfaces ? $\endgroup$ – Alex Degtyarev May 24 '14 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 May 24 '14 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 May 24 '14 at 10:29
  • 2
    $\begingroup$ Rigid Calabi--Yau threefolds (e.g. arxiv.org/abs/1102.1854) are odd dimensional examples. $\endgroup$ – Sasha May 24 '14 at 10:30
  • 1
    $\begingroup$ 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 May 24 '14 at 10:33
4
$\begingroup$

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

$\endgroup$
  • $\begingroup$ 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 May 24 '14 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. $\endgroup$ – F_L May 24 '14 at 14:17

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.