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!



Two approaches to compute the signature of a Kaehler manifold
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.


