  We prove bounds on the number of incidences between a set of algebraic curves
in $\mathbb{C}^2$ and a Cartesian product $A\times B$ with finite sets
$A,B\subset \mathbb{C}$. Similar bounds are known under various conditions, but
we show that the Cartesian product assumption leads to a simpler proof. This
assumption holds in a number of interesting applications, and with our bound
these applications can be extended from $\mathbb{R}$ to $\mathbb{C}$. The proof
is a new application of the polynomial partitioning technique introduced by
Guth and Katz.
