Finite-dimensionality of the cohomology of a divisor on a compact Riemann surface

For $E={\mathcal{O}}_X(D)$, the degree-zero Dolbeault cohomology is the kernel of ${\bar{\mathrm{\partial }}}_E$, hence the holomorphic-section space; it is finite-dimensional and equals $H^{0,0}(E)$ by [F7]. On a curve there are no $(0,2)$-forms, so every smooth $(0,1)$-form is ${\bar{\mathrm{\partial }}}_E$-closed and degree-one Dolbeault cohomology is exactly the displayed quotient. By [F7] this quotient is finite-dimensional and every class has exactly one harmonic representative.

The canonical comparison of [F8] identifies the degree-zero and degree-one Dolbeault groups with $H^0(X,{\mathcal{O}}_X(D))$ and $H^1(X,{\mathcal{O}}_X(D))$, respectively, so both sheaf-cohomology spaces are finite-dimensional. By [F9], ${\check{H}}^1(\mathfrak{U},{\mathcal{O}}_X(E))$ is isomorphic to $H^1(X,{\mathcal{O}}_X(E))$ whenever a finite good cover subordinate to holomorphic frame domains is supplied. By [F4], $H^0(X,{\mathcal{O}}_X(D))\cong L(D)$. Thus $\mathrm{\ell }(D)$ and $i(D)$ are finite nonnegative integers and their difference $\chi ({\mathcal{O}}_X(D))$ is an integer; no nonnegativity of that difference is asserted.

Suppose $D\sim D^{\mathrm{\prime }}$. By [F3] there is a nonzero meromorphic function $u$ with $(u)=D^{\mathrm{\prime }}-D$. The isomorphism in [F5] identifies the sheaves of holomorphic sections of ${\mathcal{O}}_X(D)$ and ${\mathcal{O}}_X(D^{\mathrm{\prime }})$, so [F2] induces an isomorphism on $H^1$; on global sections the map is multiplication by $1\mathrm{/}u$. Thus both dimensions $\mathrm{\ell }$ and $i$ are unchanged, and their difference $\chi$ is unchanged as well.