  We study resonances associated to Schr\"odinger operators with compactly
supported potentials on ${\mathbb R}^d$, $d\geq3$, odd. We consider compactly
supported potentials depending holomorphically on a complex parameter $z$. For
certain such families, for all $z$ except those in a pluripolar set, the
associated resonance-counting function has order of growth $d$. Our proofs use
some results from several complex variables.
