I am sorry (I phrased the question wrongly). I edited my question. I mean why is the Hodge conjecture equivalent to "Rational cohomology is generated by Chern classes of holomorphic vector bundles".

Einstein summation implied above. Also $\frac{\partial}{\partial z} = \frac{\partial}{\partial x} - i\frac{\partial}{\partial y}$

Dennis, $\partial (\omega _{IJ} dz^{I} \wedge dz^{\bar{J}} = \frac{\partial \omega _{IJ}}{\partial z^k} dz^k \wedge dz^{I} \wedge dz^{\bar{J}}$ and $\bar{\partial} (\omega _{IJ} dz^{I} \wedge dz^{\bar{J}} = \frac{\partial \omega _{IJ}}{\partial \bar{z}^k} d\bar{z}^k \wedge dz^{I} \wedge dz^{\bar{J}}$

@ Daniel, I don't know and actually part of my question is that. @ Donu, I am sorry, my background is more in the analytic side, so does $Ext ^1 (\mathcal{E}, ker f) = 0$ follow obviously from Cartan? (I mean is their an injective resolution of Coherent sheaves?). I apologise for the silly question.

By Hormander, I meant you can use a version of it to extend sections and thus essentially prove a special case of Cartan anyway :).