Yesterday I asked the following question to which abx has given a positive answer.

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

But I suddenly realized that those Fano manifolds I mentioned in the quesion above (flag manifolds $G/P$ and Fano contact manifolds) not only have trivial odd Betti numbers but have stronger properties: their Hodge numbers satisfy $h^{p,q}=0$ whenever $p\neq q$.

But the example abx mentioned (some smooth complete intersections) and the K3 surfaces seem not to support the above property.

So I would like to modify my original question by asking the following: does there exist a compact Kähler manifold whose first Chern class is trivial such that its Hodge numbers satisfy $h^{p,q}=0$ whenever $p\neq q$?

  • 6
    $\begingroup$ It depends on what do you mean by vanishing of the first Chern class. On one hand, Enriques surfaces have $c_1 = 0$ in cohomology with $\mathbb{Q}$ coefficients and its Hodge diamond is diagonal. On the other hand, if the canonical class of $X$ is trivial then $h^{n,0}(X) \ne 0$. $\endgroup$
    – Sasha
    May 25 '14 at 9:01
  • $\begingroup$ @Sasha What I mean is $c_1=0$ in $H^{2}(M;\mathbb{R})$. I want to know if, for any $n$, there exists an $n$-dimensional example. $\endgroup$
    – Kevin
    May 25 '14 at 11:37
  • $\begingroup$ @Kevin: In that case Sasha's examples are valid -- you are killing the torsion in cohomology when you use $\mathbb{R}$-coefficients. So there are Enriques surfaces, and many more examples in dimensions $>2$. $\endgroup$ May 25 '14 at 13:11

If the Chern class vanishes over integers, it's Calabi-Yau manifold, and it has a holomorphic (n,0)-form by Bogomolov's theorem (see there: Two definitions of Calabi-Yau manifolds). If you relax your condition by asking $c_1$ to vanish over reals, then there are many examples, such as an Enriques surface; its only non-zero Hodge numbers are $h^{1,1}=10$, $h^{0,0}=1$, $h^{2,2}=1$.

The quotients of Calabi-Yau manifolds often satisfy $h^{p,q}=0$ for all $p\neq q$. However, there is an interesting question (still unresolved, as far as I know): does there exist a quotient of a compact torus which is Kahler and has this property?


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.