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The complex projective line is isomorphic to the 2-sphere, and so, has genus $0$. Does this result for all $CP^N$, that is, is the genus of $CP^N$ equal to $0$, for all $N$?

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The geometric genus (the dimension of the space of global sections of the canonical sheaf) of projective $n$-space is zero. See Hartshorne II.8.

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Talking about the arithmetic genus (http://en.wikipedia.org/wiki/Arithmetic_genus), it's the alternating sum of Hodge numbers all of which are 0. So, in short, yes.

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