known to also hold in the complex plane $\C^2$ [@Toth; @Zahl]. In recent years there has been a growing interest in high dimensional variants of line-point incidence bounds [@SSZ13; @Kollar14; @Rudnev14; @SS14; @ST12; @BS14]. This is largely due to the breakthrough results of Guth and Katz [@GK10] who proved the Erdös distinct distances conjecture. One of the main ingredients in their proof was an incidence theorem for configurations of lines in $\R^3$ satisfying some ‘truly 3 dimensional’ condition (e.g, not too many lines in a plane). The intuition is that, in high dimensions, it is ‘harder’ to create many incidences between points and lines. This intuition is of course false if our configuration happens to lie in some low dimensional space. In this work we prove stronger line-point incidence bounds for sets of points that do not contain a large low dimensional subset.

To state our main theorem we first restate the Szemerédi-Trotter theorem as a bound on the number of $r$-rich lines (lines containing at least $r$ points) in a given set of points. Since our results will hold over the complex numbers we will switch now from $\R$ to $\C$. The complex version of Szemeredi-Trotter was first proved 