# Incidences in the complex plane

Recently Joshua Zahl and I have uploaded to arXiv a paper about incidence in ${\mathbb C}^2$. In this post I briefly survey what was previously known concerning such incidences, and then describe what the new paper is about.

Let $\Gamma$ be a set of curves and let ${\cal P}$ be a set of points, both in ${\mathbb F}^2$ for some field $\mathbb F$. We say that the arrangement $({\cal P},\Gamma)$ has $k$ degrees of freedom and multiplicity-type $s$ if

• For any $k$ points from $\cal P$, there are at most $s$ curves from $\Gamma$ that contain all of them.
• Any pair of curves from $\Gamma$ intersect in at most $s$ points.
The following theorem is currently the best known general incidence bound in ${\mathbb R}^2$ (better bounds are known for some special cases, such as circles and parabolas). For a proof, see for example this paper by Pach and Sharir.

Theorem 1. Let $\cal P$ be a set of $m$ points and let $\Gamma$ be a set of $n$ algebraic curves of degree at most $D$, both in ${\mathbb R}^2$. Suppose that $({\cal P},\Gamma)$ has $k$ degrees of freedom and multiplicity type $s$. Then $I({\cal P},\Gamma) = O_{k,s,D}\big(m^{\frac{k}{2k-1}}n^{\frac{2k-2}{2k-1}}+m+n\big).$

In the complex plane, things progressed rather slowly. Csaba Tóth derived a matching incidence bound for the special case of complex lines in ${\mathbb C}^2$ (i.e., the Szemerédi–Trotter bound). This is the longest case of refereeing that I know of, and if I recall correctly it took 13 years (!!!) for the journal to accept this paper (and it is still not published). In the meantime, a similar result was independently obtained by Solymosi and Tao, by using the polynomial method.

Solymosi and Tao considered ${\mathbb C}^2$ as ${\mathbb R}^4$, obtaining an incidence problem between points and two-dimensional surfaces. By introducing the method of “constant sized” polynomial partitioning, they derived the bound $O(m^{2/3+\varepsilon}n^{2/3}+n+m)$ for such a point-surface incidence problem in ${\mathbb R}^4$, for any $\varepsilon>0$. However, this bound applies only when the two-dimensional surfaces intersect transversely (that is, at any intersection point of two such surfaces, the two tangent planes intersect in a single point). This is indeed the case for complex lines, which turn into two-dimensional planes in ${\mathbb R}^4$ that intersect transversely. This bound also applies to complex unit circles in ${\mathbb C}^2$. Although the corresponding two-dimensional real surfaces in ${\mathbb R}^4$ do not intersect transversely, one can cut each complex circle into a constant number of pieces and then treat different types of pieces separately (i.e., consider a constant number of incidence problems in ${\mathbb R}^4$, each with $n$ identical pieces). Unfortunately, as far as I know, no other interesting example of complex algebraic curves that satisfy the transversality property is known.

The next step forward was by Zahl, who showed how to remove the $\varepsilon$ from the bound of Solymosi and Tao. So at that point, it was known how to obtain the same bound of Theorem 1 for the cases of complex lines and of complex unit circles. Recently, Solymosi and de Zeeuw derived another incidence bound in ${\mathbb C}^2$. This result obtains the bound of Theorem 1 for any type of curves, but only for the special case where the point set is a Cartesian product (that is, of the form $A\times B$ for some $A,B \subset {\mathbb R}$). While this might sound as a rather special case, it is actually a common case and several other papers already rely on this result. The paper only describes the case of two degrees of freedom, but Frank de Zeeuw tells me that it can easily be extended to $k$ degrees of freedom.

Another two recent related results are a paper by Dvir and Gopi and a paper by Zahl. Both of these papers study incidences with lines in ${\mathbb C}^d$ when no too many points are on a common $\ell$-flat (an $\ell$-dimensional affine subspace).

In the new work with Zahl, we derive a significantly more general incidence bound in the complex plane, which matches Theorem 1 up to an extra $\varepsilon$ in the exponent.

Theorem 2. Let $\cal P$ be a set of $m$ points in ${\mathbb C}^2$ and let $\Gamma$ be a set of $n$ algebraic curves of degree at most $D$. Suppose that $({\cal P},\Gamma)$ has $k$ degrees of freedom and multiplicity type $s$. Then for any $\varepsilon>0$, we have $I({\cal P},\Gamma) = O_{k,s,D}\big(m^{\frac{k}{2k-1}+\varepsilon}n^{\frac{2k-2}{2k-1}}+m+n\big).$

To obtain this bound, we begin in the same way as Solymosi and Tao. That is, we consider ${\mathbb C}^2$ as ${\mathbb R}^4$, and apply a constant-sized polynomial partitioning. Handling the incidences in the cells is straightforward. Moreover, handling surfaces that intersect the partitioning in a zero- or one-dimensional variety is also standard. The difficult part is handling incidences with surfaces that are fully contained in the partitioning. To handle this part, Solymosi and Tao relied on the transversality restriction.

Instead of relying on transversality, we rely on the fact that our surfaces originated from complex curves. When two complex curves in ${\mathbb C}^2$ intersect, their tangent lines at the intersection point are either identical or intersect in a single point. Thus, when a pair of our two-dimensional surfaces in ${\mathbb R}^4$ intersect, their tangent planes at that point are either identical or intersect at a single point. To make this argument rigorous, we rely on the Cauchy–Riemann equations.

Next, recall that we are only interested in two-dimensional surfaces that are fully contained in our polynomial partitioning, which is a three-dimensional surface $Z \subset {\mathbb R}^4$. When two such surfaces $S,S'$ intersect in a regular point of $Z$, their tangent planes are contained in the tangent hyperplane of $Z$ at that point. In such an intersection, the tangent planes of $S$ and $S'$ cannot intersect at a single point, since then they would not be contained in a common hyperplane. Thus, if $S$ and $S'$ intersect at a regular point of $Z$, their tangent planes coincide at that point.

We use the coinciding tangent planes property to show that the set of complex curves in ${\mathbb C}^2$ that are fully contained in $Z$ (when considered as two-dimensional surfaces in ${\mathbb R}^4$) correspond to a foliation of $Z$. To show that, we use Frobenius’ theorem on integrable distributions. This implies that every regular point of $Z$ is contained in at most one of our two-dimensional surfaces. Since the singular points of $Z$ can be handled with standard techniques, this solves the aforementioned difficulties.

I should state that the proof is significantly more involved, and that in my above description I swept under the rug several issues. For example, in addition to removing the singular points of $Z$, we also need to remove “bad” regular points. We use Chevalley’s upper semi-continuity theorem to show that there are not too many such bad points. There are several other issues, but in this post I only wanted to give a general high-level idea of how the proof works.