EQUIVARIANT RIEMANN–ROCH FOR SURFACES VIA RESIDUE FORMULAS

Let $X$ be a smooth projective surface defined over $\CC$, equipped with a finite group action $G$. A line bundle $L$ on $X$ is called $G$-equivariant, if the $G$-action on $X$ extends to a compatible linear action on $L$. This extends to an action on its sheaf of sections, $\LL$, which in turn induces an action on the cohomology groups $H^0(X,\LL)$, $H^1(X,\LL)$ and $H^2(X,\LL)$, and refines them into $G$-representations. The virtual Euler characteristic, $\chi_G(X,\LL)$, is defined as the character of the formal alternating sum $[H^0(X, \LL)]-[H^1(X, \LL)]+[H^2(X, \LL)]$, considered as an element (a virtual representation) in the representation ring of $G$. The equivariant Riemann--Roch aims to compute this characteristic using geometric data. Donu Arapura gave an explicit formula in the case where $X$ is a smooth projective curve, using logarithmic Gauss--Manin connections and residue theorems. This thesis examines the viability of Arapura's methods in computing the equivariant characteristic of line bundles on smooth projective surfaces.