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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.

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3 Answers 3

up vote 7 down vote accepted

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 $Td(S)=1/12 (K^2+\chi_{top}(S))$

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Why did you say what $p_1$ and $L$ are and you didn't with remaining six different symbols? Nice answer, though. –  Mauricio Mar 1 '12 at 22:55
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Ok, let me explain myself a little more. $K_s$ is the canonical divisor of $S$: second exterior power of $T^*S$. Then, the second line above says, the relation you asked for is: "the self intersection of the canonical -2topological Euler characteristic is 3 times the signature". Now, going a little further, I included at the very last equation, $3(L+c_2(s))=12 Td (S)$ in order to include the relation among the classes $L$, $Td$ and the Chern classes. (This last thing was just a bonus) –  Csar Lozano Huerta Mar 2 '12 at 2:30

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.

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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.

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I found your lecture notes very useful. Thank you for bringing them to my attention. By the way, I think that on page 91 there is a typo at the very bottom. You didn't use $\epsilon$ to define the regions $D^{\pm}$. –  Mauricio May 3 '12 at 23:46
    
I'm glad that the notes helped. Thanks for the correction. –  Liviu Nicolaescu May 4 '12 at 12:20

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