# Relation between Chern characteristic and Pontryagin characteristic

A 2-dim complex manifold can be viewed as a 4-dim real manifold. What is the relation between the Chern characteristic and the Pontryagin characteristic of the tangent bundle? It should be $p_1=n_1 c_1^2 + n_2 c_2$. What are the values of $n_1$ and $n_2$?

$p_1=c_1^2-2c_2$. This immediately follows from Theorem 4.5.1 in the book "Topological methods in algebraic geometry" by F. Hirzebruch.
• Thanks! From Theorem 4.5.1, in general we have $1-p_1+p_2-p_3+\cdots =( 1-c_1+c_2-c_3 + \cdots)( 1+c_1+c_2+c_3 + \cdots)$. May 8 '14 at 16:43