Oscillation and boundary curvature of holomorphic curves in Cn

The number of isolated intersections between a smooth curve in Euclidean space and an arbitrary hyperplane can be majorized by a weighted sum of integral Frenet curvatures of the curve. In the complex Hermitian space one can derive a similar result for holomorphic curves but with much better weights. The proof of this result is based on a generalization of the Milnor-Fary theorem for complex Hermitian spaces: the expected integral curvature of a random hyperplanar Hermitian orthogonal projection of a smooth curve in C-n is equal to the integral curvature of the projected curve itself. In the appendix we show how this technique allows one to improve the known estimates for real analytic curves in Euclidean space.