The first chiral homology group in higher genus

We extend the theory of the first chiral homology group of vertex algebras, developed by van Ekeren and Heluani for elliptic curves, to compact Riemann surfaces of arbitrary genus. Our approach realizes a genus $g$ surface by iterated self-sewing of $g$ handles onto the Riemann sphere, each governed by a sewing parameter $\rho_i$ in a punctured disc, so that the construction of van Ekeren-Heluani is recovered. We construct an explicit complex computing the chiral homology groups $H^{\mathrm{ch}}_0$ and $H^{\mathrm{ch}}_1$ of a vertex algebra $V$ on a genus $g$ surface with $n$ marked points, equip it with a projectively flat connection, with an explicit central-charge anomaly, over the $g$-dimensional space of sewing parameters, and prove a genus $g$ Fourier-space Borcherds identity for the associated modified vertex operators. We show that the same two finiteness hypotheses isolated by van Ekeren and Heluani in genus $1$ - finite dimensionality of the first Poisson homology $\HP_1(R_V)$ of the Zhu $C_2$- algebra, and finite generation of a certain Koszul homology of the associated graded algebra - imply finite dimensionality of $H^{\mathrm{ch}}_1(X,V)$ for every genus $g$ and every vertex algebra $V$, answering a question left open in their work. Using the degeneration $\rho_i \to 0$ together with the factorization theorem of Damiolini- Gibney-Tarasca, we relate the totally degenerate limit of $H_1^{\mathrm{ch}}$ to the Hochschild homology of an iterated construction on the Zhu algebra, and deduce vanishing of the first chiral homology group in every genus for the same classically free, rational vertex algebras treated in genus one.