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.



Geometric meaning of L-genus
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 $Td(S)=1/12 (K^2+\chi\_{top}(S))$ 


