Point-Curve Incidences in the Complex Plane

We prove an incidence theorem for points and curves in the complex plane. Given a set of mpoints in ℝ^2 and a set of n curves with k degrees of freedom, Pach and Sharir proved that the number of point-curve incidences is O(m k/2k−1 n 2k−2/2k−1+m+n). We establish the slightly weaker bound O_ε(m k/2k−1 + ε n 2k−2/2k−1+m+n) on the number of incidences between m points and n (complex) algebraic curves in ℂ^2 with k degrees of freedom. We combine tools from algebraic geometry and differential geometry to prove a key technical lemma that controls the number of complex curves that can be contained inside a real hypersurface. This lemma may be of independent interest to other researchers proving incidence theorems over ℂ