The Szemeredi-Trotter Theorem in the Complex Plane

It is shown that $n$ points and $e$ lines in the complex Euclidean plane ${\mathbb{C}}^2$ determine $O(n^{2\mathrm{/}3}{e}^{2\mathrm{/}3}+n+e)$ point-line incidences. This bound is the best possible, and it generalizes the celebrated theorem by Szemer'edi and Trotter about point-line incidences in the real Euclidean plane ${\mathbb{R}}^2$.