# Arithmetic and geometric genus

There are two notion of genus in algebraic geometry, namely arithmetic genus $p_a=(-1)^{\dim X}(\chi(\mathcal{O}_X)-1)$ and geometric genus $p_g=\dim H^0(X,\Omega^{\dim X})$. I keep forgetting definition of these, or being confused which is which. Are there any good ways to remember them?

More precisely I would like to associate these definition with these names "arithmetic" and "geometric".

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You might look at this answer – Georges Elencwajg Jan 4 '13 at 21:19
As a sort of mnemonic, I tell myself that geometric genus counts something geometric, like holes or independant loops or independant cycles and is thus always positive or zero, whereas arithmetic genus has some pleasant formal properties (which is why Hirzebruch introduced them) but has no geometric interpretation and in particular may even be negative. – Georges Elencwajg Jan 4 '13 at 21:30
Thank you for the comment. – M. K. Jan 5 '13 at 8:56