Equivariant Riemann-Roch for Surfaces via Connections and Residues

Let $X$ be a smooth projective surface over $\mathbb{C}$ with an effective action of a finite group $G$, and let $L$ be a $G$-linearized line bundle. We develop a residue-theoretic and birational approach to computing the equivariant Euler characteristic $χ_G(X,L)$. Every such $L$ admits an equivari