  We study the 2+1 dimensional XY model at nonzero chemical potential $\mu$ on
deformed integration manifolds, with the aim of alleviating its sign problem.
We investigate several proposals for the deformations, and considerably improve
on the severity of the sign problem with respect to standard reweighting
approaches. We present numerical evidence that the reduction of the sign
problem is exponential both in $\mu^2$ and in the spatial volume. We also
present a new approach to the optimization procedure based on reweighting, that
sensibly reduces its computational cost.
