Moving subvarieties by endomorphisms (Q2480898)

The main result of the author is as follows. Let \(A\) be a semi-abelian variety over an algebraically closed field \(K\) of characteristic \(0\), and \({\text{ End}}(A)\) the endomorphism group of \(A\). Then for every subvariety \(X\subset A\) of strictly smaller dimension, the set \(\bigcup_{f\in{\text{ End}}(A)} f(X(K))\) is strictly smaller than \(A(K)\). In the case that \(K={\mathbb{C}}\) this follows easily by endowing \(A({\mathbb{C}})\) with a measure such that for every \(f\in {\text{ End}}(A)\) the set \(f(X({\mathbb{C}}))\) has measure \(0\), and using that \({\text{ End}}(A)\) is countable. When \(K\) is a countable field, for instance when \(K\) is the algebraic closure of \({\mathbb{Q}}\), this result is new. The main ingredients in the author's proof are the following. First he uses the fact, proved by himself, that for every positive integer \(r\) there is a finitely generated subgroup \(\Gamma\) of \(A(K)\), such that \(\Gamma\cap B(K)=\{ 0\}\) for every algebraic subgroup \(B\) of \(A\) of strictly smaller dimension. Second he uses the Mordell-Lang conjecture, which is now a theorem thanks to work of Laurent, Hindry, Vojta, Faltings and McQuillan: for any subvariety \(X\) of \(A\) and any subgroup \(\Gamma\) of \(A(K)\) of finite rank, the intersection \(X(K)\cap\Gamma\) is a union of finitely many translates of algebraic subgroups of \(A\) which are all contained in \(X\).