  In this technical note we introduce a manifestly gauge-invariant and
supersymmetric procedure to regularize and renormalize one-loop divergences of
chiral multiplets in two-dimensional N=(2,2) theories in curved spacetime. We
apply the method, a version of Pauli-Villars regularization, to known examples
of supersymmetric localization and demonstrate that the partition functions are
renormalized. An accompanying ancillary file, a Mathematica notebook,
implements the relevant supergravity transformations.
