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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 $n_2$?

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$p_1=c_1^2-2c_2$. This immediately follows from Theorem 4.5.1 in the book "Topological methods in algebraic geometry" by F. Hirzebruch.

This identity holds for any complex vector bundle of any rank over any reasonable space.

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    $\begingroup$ Thanks! From Theorem 4.5.1, in general we have $1-p_1+p_2-p_3+\cdots =( 1-c_1+c_2-c_3 + \cdots)( 1+c_1+c_2+c_3 + \cdots)$. $\endgroup$ May 8 '14 at 16:43

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