Stokes type-formula

Question

Let $X$ be a complex manifold. Is the following identity true?

$$\int_X\overline {\partial} \alpha=\int_{\partial X} \alpha$$ where $\alpha $ is a differential form on $X$, and $\partial X $ is the boundary of $X$.

The same question if we replace $\overline {\partial} \alpha $ by $\partial \alpha $?

Answer 1

Let $X$ be the unit disk in the complex plane, $\alpha=-z\,d\bar{z}$, so $\bar\partial\alpha=0$, but $\int_{\partial X}\alpha=2\pi i$. Conjugate this example to get an example for $\partial$.