I am struggling to understand a passage from Claire Voisin’s book on Hodge Theory.
In page 53, at the beginning of the section 2.3.1, there is the following assertion:
“…the bundle $\Omega_X^{1,0}$ of complex differential forms of type 1,0, i.e. $\mathbb C$-linear forms…”
I don’t see why forms of type 1,0 should be C-linear.
Maybe I don’t understand correctly what $\Omega_X^{1,0}$ is. To me it is supposed to be $\hom_\mathbb R(T^{1,0}_X, \mathbb C)$.
Am I wrong here? If not, why is $\hom_\mathbb R(T^{1,0}_X, \mathbb C)$ equal to $\hom_\mathbb C(T^{1,0}_X, \mathbb C)$?
UPDATE: In Page 64/65 of the same book the author mentions the same result but with vectors spaces instead of vector bundles, and there is explicit that $\hom_\mathbb R (V,\mathbb C)$ splits as a sum of $\mathbb C$-linear forms and $\mathbb C$-antilinear forms, that correspond to the forms of type 1,0; 0,1.
