Explicitly Understanding a Case of Dolbeault's Theorem.

Question

If $\Omega^p$ is the sheaf of holomorphic $p$-forms on a complex manifold $M$, then Dolbeault's theorem states $$ H^{p,q}(M) \cong H^q(M,\Omega^p) $$ Setting $p=0,q=1$, we get $$ H^{0,1}(M) \cong H^1(M,\cal O) \cong \check{H}^1(\cal O) $$ where $\cal O$ is the structure sheaf, and $\check{H}$ is Cech cohomology. Is there an explicit way to see the isomorphism between Cech cohomology and Dolbeault cohomology in this case?

Answer 1

Consider a class $[\alpha]\in H^{0,1}(M)$, we can find a fine enough open cover $\{U_i\}_{i\in I}$ so that $\alpha$ is locally $\bar\partial$-exact, so $\alpha|_{U_i}=\bar\partial \beta_i$ for some smooth functions $\beta_i$. Notice that on the intersection $U_i\cap U_j$, $\bar\partial(\beta_i-\beta_j)=0$, so $\beta_{ij}=\beta_i-\beta_j$ is holomorphic on $U_i\cap U_j$. You can verify that $U_{ij}$ defines a Čech 1-cocycle, this gives a Čech cohomology class. This does not depend on the choice of representative $\alpha$, because if we choose $\alpha +\bar\partial \gamma$ instead each $\beta_i$ will be changed to $\beta_i+\gamma$ and $\beta_{ij}$ will be unchanged.

For the reverse direction, by Leray's theorem the Čech cohomology can be represented by Čech cocycle in a Leray cover. Fix one such cover and a Čech 1-cocyle $\{f_{ij}\in \mathcal O(U_i\cap U_j\}_{i,j\in I}$, it is always possible to find smooth functions $f_i\in C^{\infty}(U_i)$ so that $f_{ij}=f_i-f_j$ due to partition of unity. (See for example, Griffiths-Harris.) Then $\{\bar\partial f_i\}$ glues together to give a $(0,1)$-form. This again is independent on the choice of $f_i$'s, if $f_{ij}=f_i-f_j=g_i-g_j$, then $f_i-g_i=f_j-g_j$ so $\{f_i-g_i\}_{i\in I}$ glues to a global function.