The residue pairing for line-bundle cohomology

By the global comparison in [F2], $H^1(X,{\mathcal{O}}_X(D))$ is canonically identified with the quotient of smooth $E$-valued $(0,1)$-forms by ${\bar{\mathrm{\partial }}}_E$ of smooth sections. By [F3], the first variable is finite-dimensional. For $\xi$ and $\omega$, choose any representative $\theta$ of $\xi$ in this quotient and define $B$ by the displayed integral. Evaluation $E\otimes E^{\ast }\to \mathbb{C}$ makes the integrand a smooth top-degree form, and the integral is complex-bilinear in $\theta$ and $\omega$. Thus it defines a bilinear expression on representatives.

If ${\theta }^{\mathrm{\prime }}=\theta +{\bar{\mathrm{\partial }}}_{E}f$ for a smooth section $f$ of $E$, then [F7] and holomorphy of $\omega$ give $({\theta }^{\mathrm{\prime }}-\theta )\wedge \omega =\bar{\mathrm{\partial }}(f\omega )$. On a curve the $(2,0)$ component of $d(f\omega )$ vanishes, so $\bar{\mathrm{\partial }}(f\omega )=d(f\omega )$. The exact-form integral is zero by [F12]. Thus the integral is independent of $\theta$. The global comparison in [F2] is canonical and evaluation is frame-independent. If a fixed-cover class is used, its refinement-compatible comparison gives the same sheaf class, so the pairing is independent of that supplied cover as well.

Let $c=(c_{ij})$ and $({\eta }_i)$ be as in statement 3 and put ${\mu }_i:={\eta }_i\tilde{\omega }$. By [F4], the scalar representative satisfies $g_{ij}={\eta }_j-{\eta }_i$ with the ordered Čech sign. The meromorphic sections ${\eta }_i{s}_D$ of $E$ satisfy $({\eta }_j{s}_D)-({\eta }_i{s}_D)=g_{ij}{s}_D=c_{ij}$, which is holomorphic by [F5]. Evaluating against the holomorphic $E^{\ast }$-valued form $\omega$ shows ${\mu }_j-{\mu }_i=g_{ij}\tilde{\omega }$ is a holomorphic differential. Therefore the ${\mu }_i$ have identical principal parts and residues wherever their domains overlap. Define $P$ to be the set of points at which one (equivalently every) local ${\mu }_i$, with $p\in U_i$, has a pole; the equivalence follows from the holomorphic differences. Given $p\in X$, choose $i$ with $p\in U_i$ and a coordinate disk $V$ about $p$ with compact closure contained in $U_i$. The meromorphic differential ${\mu }_i$ has finitely many poles in $\bar{V}$, and the pole sets agree on overlaps, so $P\cap V$ is finite. Thus $P$ is locally finite; compactness of $X$ makes $P$ finite, and the residue sum in statement 3 is well defined.

Choose a smooth partition of unity $({\rho }_i)$ subordinate to $\mathfrak{U}$ by [F10] and set $\mu ={\sum }_i{\rho }_i{\mu }_i$ on $X\setminus P$, where $P$ is the finite pole set from step 2.2. For $x\in U_i$, define $u_i:={\sum }_j{\rho }_j({\mu }_i-{\mu }_j)$, with each summand taken on $U_i\cap U_j$ and extended by zero off $U_j$. This extension is smooth because $\mathrm{supp}({\rho }_j)\subseteq U_j$, while the difference ${\mu }_i-{\mu }_j$ is holomorphic on the overlap by step 2.2. Hence $u_i$ is smooth across $P$; on $U_i\setminus P$, using ${\sum }_j{\rho }_j=1$ gives $u_i={\mu }_i-\mu$. Also $u_j-u_i={\mu }_j-{\mu }_i$, so the forms $\bar{\mathrm{\partial }}{u}_i$ glue to a smooth $K$-valued $(0,1)$-form on $X$ representing the Dolbeault class of the product Čech cocycle $(g_{ij}\tilde{\omega })$ in $H^1(X,K)$. By naturality of [F2], this product class is the image of $\xi$ multiplied by $\omega$, so $B(\xi ,\omega )=(2\pi i{)}^{-1}{\int }_X\bar{\mathrm{\partial }}{u}_i$. On $X\setminus P$, $\bar{\mathrm{\partial }}{u}_i=-\bar{\mathrm{\partial }}\mu$ and $d\mu =\bar{\mathrm{\partial }}\mu$, since $\mu$ has type $(1,0)$ on a curve. Remove disjoint coordinate disks $B_{ϵ}(p)$ around $P$ and apply [F11] to get ${\int }_{X\setminus {\bigcup }_p\mathrm{int}{B}_{ϵ}(p)}\bar{\mathrm{\partial }}\mu ={\int }_{\mathrm{\partial }(X\setminus {\bigcup }_p\mathrm{int}{B}_{ϵ}(p))}\mu =-{\sum }_{p\in P}{\int }_{\mathrm{\partial }{B}_{ϵ}(p)}\mu$. If $P=\mathrm{\varnothing }$, this is Stokes on all of $X$ with empty boundary and gives zero, matching the empty residue sum. Near each $p$, $\mu -{\mu }_i$ is smooth and bounded, so its integral around $\mathrm{\partial }{B}_{ϵ}(p)$ tends to zero, while the integral of ${\mu }_i$ tends to $2\pi i{\mathrm{Res}}_p({\mu }_i)$. The smooth form $\bar{\mathrm{\partial }}{u}_i$ extends across $P$, so its integral over the removed disks tends to zero as well. Taking $ϵ\downarrow 0$ yields $(2\pi i{)}^{-1}{\int }_X\bar{\mathrm{\partial }}{u}_i={\sum }_{p\in P}{\mathrm{Res}}_p({\mu }_i)$, which is the residue formula in statement 3. This also derives the sign from the positive boundary orientation of each deleted disk and the ordered Čech differential in [F4].