  We introduce a simple procedure to integrate differential forms with
arbitrary holomorphic poles on Riemann surfaces. It gives rise to an intrinsic
regularization of such singular integrals in terms of the underlying conformal
geometry. Applied to products of Riemann surfaces, this regularization scheme
establishes an analytic theory for integrals over configuration spaces,
including Feynman graph integrals arising from two dimensional chiral quantum
field theories. Specializing to elliptic curves, we show such regularized graph
integrals are almost-holomorphic modular forms that geometrically provide
modular completions of the corresponding ordered $A$-cycle integrals. This
leads to a simple geometric proof of the mixed-weight quasi-modularity of
ordered A-cycle integrals, as well as novel combinatorial formulae for all the
components of different weights.
