TITLE: Geometric meaning of L-genus
QUESTION [16 upvotes]: 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.

REPLY [13 votes]: 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.