Two approaches to compute the signature of a Kaehler manifold - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T14:05:48Z http://mathoverflow.net/feeds/question/93522 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/93522/two-approaches-to-compute-the-signature-of-a-kaehler-manifold Two approaches to compute the signature of a Kaehler manifold Zhang Xiao 2012-04-08T22:06:27Z 2013-01-08T18:33:51Z <p>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 $H_{2n}(M,\mathbb{R})$:</p> <p>1.by Hodge index theorem $\sigma(M)=\sum_{p,q}(-1)^p h^{p,q}$, here $h^{p,q}$ stands for the Hodge numbers.</p> <p>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.</p> <p>My questions are</p> <p>1.Since these two approaches rest on different levels of cohomology theory, how are they interrelated? </p> <p>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 <em>Neue topologische Methoden</em>. However, I am wondering if someone could relate these two approaches on a more fundamental level. </p> <p>To be precise,</p> <p>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. </p> <p>Or, does anyone know such counterexamples that two Kaehler manifolds(notably, Kaehler surfaces, I guess) have the same Hodge numbers but different Chern numbers?</p> <p>Many thanks!</p> http://mathoverflow.net/questions/93522/two-approaches-to-compute-the-signature-of-a-kaehler-manifold/93534#93534 Answer by YangMills for Two approaches to compute the signature of a Kaehler manifold YangMills 2012-04-09T00:23:12Z 2012-04-09T00:23:12Z <p>About your last question, a recent theorem of Kotschick-Schreieder (see <a href="http://arxiv.org/abs/1202.2676" rel="nofollow">http://arxiv.org/abs/1202.2676</a> page 2) says that a linear combination of Hodge numbers equals a linear combination of Chern numbers for all projective manifolds (modulo the usual K&auml;hler symmetries) iff it is a linear combination of the numbers $\chi_p=\sum_p (-1)^q h^{p,q}$. </p> <p>Similarly, a linear combination of Hodge numbers equals a linear combination of Pontryagin numbers iff it is multiple of the signature.</p> <p>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.</p> http://mathoverflow.net/questions/93522/two-approaches-to-compute-the-signature-of-a-kaehler-manifold/118380#118380 Answer by Sergey for Two approaches to compute the signature of a Kaehler manifold Sergey 2013-01-08T18:33:51Z 2013-01-08T18:33:51Z <p>"counterexamples that two Kaehler manifolds have the same Hodge numbers but different Chern numbers?"</p> <p>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.</p> <p>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 $54$.</p>