Geometric meaning of L-genus - MathOverflow

Geometric meaning of L-genus

2012-03-01T18:13:43Z

Is there any reasonable geometric meaning of the L-genus for smooth manifolds? or perhaps easier, for complex algebraic surfaces?

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.

Answer by Csar Lozano Huerta for Geometric meaning of L-genus

2012-03-01T19:14:00Z

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.

On the other hand, cobordism theory says $p_1[S]=3\tau$ where $\tau$ is the signature of $S$.

Now (by Hodge theory)

$\tau=4\chi(\mathcal{O_S})-\chi_{top}(S)$ therefore, (pairing off with the fundamental class) the relation among $L$ and $Td$ looks like

$$K^2+\chi_{top}(S)=3\tau+3\chi_{top}(S)= 12Td(S)$$

where the second Todd class satisfies $Td(S)=1/12 (K^2+\chi_{top}(S))$

Answer by Greg Friedman for Geometric meaning of L-genus

2012-03-02T00:31:03Z

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.

Answer by Liviu Nicolaescu for Geometric meaning of L-genus

2012-03-02T11:56:09Z

I take that you ask how in the world did Hirzebruch come up with the complicated expression of the L-genus?

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

$$r^\gamma(t)=1+\sum_{n\geq 1}\gamma(\mathbb{CP}^{2n})t^{2n}.$$

In the case of signature we have

$$r^\gamma(t)=\frac{1}{1-t^2}.$$

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

$$ \xi =\log \left( \frac{1}{1-t^2}\right) \Leftrightarrow t=\tanh \xi.$$

The signature theorem follows from the above observations using a bit of algebraic combinatorics. For details see Chap. 7 of these lecture notes for a graduate course on this topic that I taught in 2008.