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Given a compact Kaehler manifold $M$ of complex dimension $2n$, there are essentially two ways to compute its signature $\sigma(M)$, i.e. the index of the intersection form on $H_{2n}(M,\mathbb{R})$:

1.by Hodge index theorem $\sigma(M)=\sum_{p,q}(-1)^p h^{p,q}$, here $h^{p,q}$ stands for the Hodge numbers.

2.by Hirzebruch signature theorem $\sigma(M)=L[M]$, here $L[M]$ stands for the $L$-genus, i.e. the characteristic number of the top $L$-class.This approach is more general since it works on any $4k$ dimensional real manifolds.

My questions are

1.Since these two approaches rest on different levels of cohomology theory, how are they interrelated?

2.Of course, one possible way to answer Question 1 is to generalize both by the Hirzebruch-Riemann-Roch on Kaehler manifolds, a point already mentioned in Hirzebruch's Neue topologische Methoden. However, I am wondering if someone could relate these two approaches on a more fundamental level.

To be precise,

Is there a formula to express the Chern numbers/Pontryagin numbers out of the Hodge numbers on a compact Kaehler manifold $M$ of complex dimension $n$? Surely it is the case for $c_n[M]$ interpreted as the Euler characteristic number.

Or, does anyone know such counterexamples that two Kaehler manifolds(notably, Kaehler surfaces, I guess) have the same Hodge numbers but different Chern numbers?

Many thanks!

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2 Answers

up vote 12 down vote accepted

About your last question, a recent theorem of Kotschick-Schreieder (see http://arxiv.org/abs/1202.2676 page 2) says that a linear combination of Hodge numbers equals a linear combination of Chern numbers for all projective manifolds (modulo the usual Kähler symmetries) iff it is a linear combination of the numbers $\chi_p=\sum_p (-1)^q h^{p,q}$.

Similarly, a linear combination of Hodge numbers equals a linear combination of Pontryagin numbers iff it is multiple of the signature.

This shows that, apart for the signature and the $\chi_p$'s, there is no universal formula to express Chern or Pontryagin numbers purely in terms of Hodge numbers. So the Hirzebruch signature formula is really an isolated phenomenon, in this sense.

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Exactly what I wondered about. Fascinating! –  Zhang Xiao Apr 9 '12 at 1:00
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"counterexamples that two Kaehler manifolds have the same Hodge numbers but different Chern numbers?"

As you explained above, Chern numbers of surfaces can be expressed in terms of the Euler number and the first Pontrjagin number, so you need dimension at least 3 for a counterexample.

In dimension 3, consider a projective space and a smooth quadric threefold. These two have the same Hodge numbers, same $c_3 = 4$ (Euler number), same $c_1 c_2 = 24$ (by Todd's theorem), but distinct degrees $c_1^3$: for the projective space it equals $64$ and for quadric it is $54$.

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