Algebraic flows on abelian varieties
type: Item
instance of: scholarly article
title: Algebraic flows on abelian varieties
zbMATH Open document ID: 1408.14140
author: Emmanuel Ullmo
author: Andrei Yafaev
published in: Journal für die Reine und Angewandte Mathematik
publication date: 2018-08-10T00:00:00Z
review text: 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…
Mathematics Subject Classification ID: 14K99
Mathematics Subject Classification ID: 28A33
Mathematics Subject Classification ID: 32Q30
Mathematics Subject Classification ID: 42B99
zbMATH DE Number: 6916481
zbMATH Keywords: Ax-Lindemann theorem
zbMATH Keywords: subtorus
zbMATH Keywords: oscillatory integral
reviewed by: G. K. Sankaran
MaRDI profile type: Publication
OpenAlex ID: W2333889975
cites work: Some Topics in Differential Algebraic Geometry I: Analytic Subgroups of Algebraic Groups
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cites work: Q3142876
cites work: Hyperbolic Ax-Lindemann theorem in the cocompact case
cites work: Q5799167
DOI: 10.1515/CRELLE-2015-0085
description: scientific article; zbMATH DE number 6916481