Algebraic flows on abelian varieties
We start with a complex abelian variety $A=\mathbb{C}^g/\Gamma$ and the associated projection map $\pi: \mathbb{C}^g\to A$. A version of the Ax-Lindemann theorem says that if $\Theta$ is a complex algebraic subvariety of $\mathbb{C}^g$, then every component of the Zariski closure of $\pi(\Theta)$ is ``weakly special''. i.e.\ a translate of an abelian subvariety. What about the closure $\overline{\pi(\Theta)}$ in the usual (analytic) topology? The conjecture is that $\overline{\pi(\Theta)}=\pi(\Theta)\cup Z_1\cup\ldots\cup Z_r$ for some \textit{real\/} weakly special subvariety of $A$. \par The authors here prove this statement in the case where $\theta=C$ is a curve. Rather than approach it directly, they treat a measure-theoretic version of the conjecture. Any subtorus $Z\subseteq A$ yields a probability measure $\mu_Z$ with support on $Z$, simply by restriction of Lebesgue measure. An algebraic subvariety $\Theta\subset\mathbb{C}^g$ of dimension $d$ also induces a measure on $\mathbb{C}^g$, and hence on $A$, for each fixed $R\in \mathbb{R}_+$, given by \[ \mu_{\Theta,R}(f)=\frac{\int_{\Theta(R)}f\omega^d}{\int_{\Theta(R)}f\omega^d}, \] where $\Theta(R)=\{\underline{z}\in \Theta\mid \max |z_k|<R\}$ and $\omega=\frac{1}{2}\sum_k dz_k\wedge d\bar{z}_k$. The conjecture then is that for some real weakly special $Z_1,\ldots Z_r$ as above and some $c_1,\dots,c_r\in\mathbb{R}_+$, the measure $\mu_{\Theta,R}$ converges weakly (i.e.\ pointwise on the space of continuous functions on $A$) to $\sum c_j\mu_{Z_j}$ as $R\to \infty$. \par In fact they prove a slightly refined version of this (always for $\Theta=C$ a curve), which identifies the $Z_j$ as Mumford-Tate tori: they are the images of the rational spans of branches of $C$ near points at infinity. \par The proofs are analytic in character, depending on estimates of oscillatory integrals, in particular the Van der Corput lemma. There is an application of Puiseux expansions to identify the Mumford-Tate tori as the relevant ones. At the end of the paper are a few illustrative examples, which give a very clear picture of what can be expected.
scientific article; zbMATH DE number 6916481