The Szemerédi-Trotter theorem in the complex plane

It is shown that n points and e lines in the complex Euclidean plane C 2 determine O(n 2/3 e 2/3+ n + e) point-line incidences. This bound is the best possible, and it generalizes the celebrated theorem by Szemerédi and Trotter about point-line incidences in the real Euclidean plane R 2 .