Geometric meaning of L-genus - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T10:18:42Z http://mathoverflow.net/feeds/question/89981 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/89981/geometric-meaning-of-l-genus Geometric meaning of L-genus Mauricio 2012-03-01T18:13:43Z 2012-05-13T10:11:11Z <p>Is there any reasonable geometric meaning of the L-genus for smooth manifolds? or perhaps easier, for complex algebraic surfaces?</p> <p>The question came up after a friend and I realized that we don't understand why one would expect to have such a formula for the signature in terms of Pontrjagin classes (i.e. the signature theorem). Any insights about this will be appreciated.</p> http://mathoverflow.net/questions/89981/geometric-meaning-of-l-genus/89986#89986 Answer by Csar Lozano Huerta for Geometric meaning of L-genus Csar Lozano Huerta 2012-03-01T19:14:00Z 2012-03-01T19:14:00Z <p>Let $S$ be a smooth algebraic complex surface. Then, there is the following relation: $$p_1=c_1(S)^2-2c_2(S)=K_S^2-2\chi_{top}(S)=3L$$ where $p_1$ is the first Pontryagin class and $L$ the L-genus.</p> <p>On the other hand, cobordism theory says $p_1[S]=3\tau$ where $\tau$ is the signature of $S$. </p> <p>Now (by Hodge theory) </p> <p>$\tau=4\chi(\mathcal{O_S})-\chi_{top}(S)$ therefore, (pairing off with the fundamental class) the relation among $L$ and $Td$ looks like</p> <p>$$K^2+\chi_{top}(S)=3\tau+3\chi_{top}(S)= 12Td(S)$$</p> <p>where the second Todd class satisfies $Td(S)=1/12 (K^2+\chi_{top}(S))$ </p> http://mathoverflow.net/questions/89981/geometric-meaning-of-l-genus/90001#90001 Answer by Greg Friedman for Geometric meaning of L-genus Greg Friedman 2012-03-02T00:31:03Z 2012-03-02T00:31:03Z <p>Hirzebruch himself has a very nice paper explaining (if I remember correctly) how he came up with the signature theorem and why the formulas arise in a fairly reasonable way. Here's the reference: MR0368023 (51 #4265) Hirzebruch, F. The signature theorem: reminiscences and recreation. Prospects in mathematics (Proc. Sympos., Princeton Univ., Princeton, N.J., 1970), pp. 3–31. Ann. of Math. Studies, No. 70, Princeton Univ. Press, Princeton, N.J. 1971.</p> http://mathoverflow.net/questions/89981/geometric-meaning-of-l-genus/90030#90030 Answer by Liviu Nicolaescu for Geometric meaning of L-genus Liviu Nicolaescu 2012-03-02T11:56:09Z 2012-05-13T10:11:11Z <p>I take that you ask how in the world did Hirzebruch come up with the complicated expression of the L-genus?</p> <p>The key fact behind this is that the signature is a genus, i.e. a ring morphism $\gamma:\Omega^\bullet_+\to\mathbb{R}$ from the oriented cobordism ring to the (ring of) reals. By Thom's work we deduce that a genus is determined by its values on $\mathbb{CP}^{2n}$ and thus by the generating series</p> <p>$$r^\gamma(t)=1+\sum_{n\geq 1}\gamma(\mathbb{CP}^{2n})t^{2n}.$$</p> <p>In the case of signature we have</p> <p>$$r^\gamma(t)=\frac{1}{1-t^2}.$$</p> <p>How does one go from this series to the the function $\frac{\xi}{\tanh \xi}$ that enters into the defintion of $L$? As a teaser, let me point out that </p> <p>$$\xi =\log \left( \frac{1}{1-t^2}\right) \Leftrightarrow t=\tanh \xi.$$</p> <p>The signature theorem follows from the above observations using a bit of algebraic combinatorics. For details see Chap. 7 of <a href="http://www.nd.edu/~lnicolae/MS.pdf" rel="nofollow">these lecture notes</a> for a graduate course on this topic that I taught in 2008. </p>