For any complex manifold $M$ we have the usual decomposition of the cotangent bundle into the direct sum $\Omega^{(1,0)} \oplus \Omega^{(0,1)}$. I know these two sub-modules as the holomorphic and anti-holomorphic forms respectively. However, I have been told that this is the terminology usually used by physicists. How to mathematicians refer to $\Omega^{(1,0)}$ and $\Omega^{(0,1)}$?

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holomorphicfor forms which are in the kernel of the $\overline\partial$ operator. See, for instance, Example 2.12 in the third edition of that book. (This used to be an answer, but I think that it is best left as a comment.) – José Figueroa-O'Farrill Dec 17 '10 at 3:00