Geometric meaning of L-genus 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.
 A: 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 $Td(S)=1/12 (K^2+\chi_{top}(S))$ 
A: 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.  
A: 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.
