Serre duality on a compact Riemann surface

Set $E={\mathcal{O}}_X(D)$ and $F_D=K_X\otimes E^{\ast }$. By [F1], the pairing $B_D$ is perfect and its two induced maps are complex-linear isomorphisms. The identification $E^{\ast }\cong {\mathcal{O}}_X(-D)$ in [F3] gives the displayed canonical twist. If a nonzero meromorphic differential $\eta$ is supplied, the local section map $h\mapsto h\eta$ gives ${\mathcal{O}}_X(K_{\eta })\cong K_X$; tensoring with ${\mathcal{O}}_X(-D)$ identifies ${\mathcal{O}}_X(K_{\eta }-D)$ with $F_D$. Hence its holomorphic sections correspond exactly to zero and the nonzero meromorphic functions $h$ satisfying $(h)+K_{\eta }-D\ge 0$, namely $L(K_{\eta }-D)$.

If $D^{\mathrm{\prime }}\le D$, the sheaf inclusion $j:{\mathcal{O}}_X(D^{\mathrm{\prime }})\hookrightarrow {\mathcal{O}}_X(D)$ induces the cohomology map $j_{\ast }$. Its dual bundle map $j^{\ast }:{\mathcal{O}}_X(D{)}^{\ast }\to {\mathcal{O}}_X(D^{\mathrm{\prime }}{)}^{\ast }$ induces $H^0(X,F_D)\to H^0(X,F_{D^{\mathrm{\prime }}})$. For a Dolbeault representative $\theta$ of $\xi \in H^1(X,{\mathcal{O}}_X(D^{\mathrm{\prime }}))$, the representative of $j_{\ast }\xi$ is $j\theta$. Evaluation satisfies $(j\theta )\wedge \omega =\theta \wedge (j^{\ast }\omega )$ pointwise, so the integral formula [F2] gives $B_D(j_{\ast }\xi ,\omega )=B_{D^{\mathrm{\prime }}}(\xi ,j^{\ast }\omega )$. Thus the stated square commutes.

By [F4], $H^1(X,E)$ is finite-dimensional; by [F1] its perfect dual is $H^0(X,F_D)$, which is therefore finite-dimensional of the same dimension. Under the identification of step 1.1, this gives $i(D)=\mathrm{\ell }(K_{\eta }-D)$ for every supplied $\eta$. If $\mathrm{deg}(K_{\eta }-D)<0$, [F3] gives $L(K_{\eta }-D)=0$; the isomorphism in [F1] then forces $H^1(X,E)=0$ as well.

If $(g)=D^{\mathrm{\prime }}-D$, [F3] gives the isomorphism $\mathrm{\Phi }:{\mathcal{O}}_X(D)\to {\mathcal{O}}_X(D^{\mathrm{\prime }})$ represented on meromorphic coefficients by $h\mapsto h\mathrm{/}g$. Its dual induces $K_X\otimes {\mathcal{O}}_X(-D^{\mathrm{\prime }})\to K_X\otimes {\mathcal{O}}_X(-D)$. Since evaluation of a section and a dual section is unchanged when they are transported by $\mathrm{\Phi }$ and its dual, the same pointwise integral calculation as in step 1.2 proves compatibility of the pairing with these isomorphisms. If ${\eta }^{\mathrm{\prime }}=g\eta$, then $K_{{\eta }^{\mathrm{\prime }}}=K_{\eta }+(g)$ and the map $h\mapsto h\mathrm{/}g$ sends $h\eta$ to $(h\mathrm{/}g){\eta }^{\mathrm{\prime }}=h\eta$; hence changing the supplied canonical divisor preserves the differential and the pairing.

Apply [F5] and [F6] to the holomorphic line bundle $G=K_X$. The comparison [F7] identifies $H^1(X,K_X)$ with its Dolbeault group, and the harmonic-star pairing identifies its complex-linear dual with $H^0(X,K_X\otimes K_X^{\ast })=H^0(X,{\mathcal{O}}_X)$. To compute the latter space, let $f\in H^0(X,{\mathcal{O}}_X)$. By [F9], $\mathrm{\mid }f\mathrm{\mid }$ attains a maximum at some point of the nonempty compact space $X$. Choose a chart about that point and a smaller coordinate disk on which the maximum is local; [F10] makes $f$ constant on that disk. The set of points having a neighborhood on which $f$ equals this constant is nonempty and open. It is closed: if $p$ is in its closure, [F12] supplies a chart at $p$; choose a connected coordinate disk inside that chart meeting the set. The identity theorem in [F10] makes $f$ equal to the same constant on the disk. Connectedness in [F11] now makes the set all of $X$. Thus every holomorphic function on $X$ is constant, and constants give $H^0(X,{\mathcal{O}}_X)\cong \mathbb{C}$, of dimension one. The duality already proved in this step then gives $h^1(X,K_X)=1$.