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Intersecting subvarieties of abelian varieties with algebraic subgroups of complementary dimension. (English) Zbl 1176.14008
Consider first an irreducible closed subvariety \(X\) of a semiabelian variety \(S\) over \({\mathbb C}\), and remove from it all translated subgroups of \(S\) that have large intersection with \(X\). The assertion in this paper is that if \(S=A\) is an abelian variety, and if \(A\) and \(X\) are both defined over \(\overline{\mathbb Q}\), then the Néron-Tate height is bounded on \(\overline{\mathbb Q}\)-points that survive this process and are in a suitable sense close to a subgroup of small dimension.
Here is a more precise statement. Let \(n=\dim S\), and for \(s\in{\mathbb N}\), define \(X^{{\text{oa,}[s]}}\) to be the complement of every subvariety \(Y\subset X\) of positive dimension such that \(Y\) is after translation also a subvariety of codimension \(<n-s\) in an algebraic subgroup of \(S\). On an abelian variety \(A\), with \(A\) and \(X\) both defined over \({\overline{\mathbb Q}}\), let \(\hat h\) denote the Néron-Tate height (with respect to some symmetric ample line bundle) and for \(\Sigma\subset A({\mathbb C})\) and \(\epsilon>0\), put \[ {\mathcal C}(\sigma,\epsilon)=\{a+b\mid a,\;b\in A({\overline{\mathbb Q}}), \;a\in\Sigma,\;\hat h(b)\leq\epsilon(1+\hat h(a))\}. \] For \(s\in{\mathbb N}\), denote by \(A^{[s]}\) the union of all algebraic subgroups of \(A\) of codimension at least  \(s\). Then the result is that for some \(\epsilon>0\) the height is bounded above on \(X^{{\text{oa,}}[s]}\cap {\mathcal C}(A^{[s]},\epsilon)\).
This result is the analogue for abelian varieties over \(\overline{\mathbb Q}\) of the bounded height conjecture of Bombieri, Masser and Zannier for \({\mathbb G}_m^n\). If \(A\) has complex multiplication there are some illuminating corollaries. Firstly, \(X^{{\text{oa,}}[s]}({\overline{\mathbb Q}})\cap A^{[1+s]}\) is finite. If, furthermore, any surjective homomorphism \(\phi: A\to B\) to an abelian variety has \(\phi(X)\) as big as possible, i.e. \(\dim \phi(X)=\min(\dim X, \dim B)\), then \(X^{{\text{oa,}}[\dim X]}\neq \emptyset\) and \(X({\mathbb Q})\cap A^{[1+\dim X]}\) is not Zariski dense in \(X\).
This last statement is a part of the following conjecture: if \(X\) is an irreducible subvariety of a semiabelian variety \(S\) (both defined over \({\mathbb C}\)), not contained in a proper subgroup of \(S\), then \(X({\mathbb C})\cap S^{[1+\dim X]}\) is not Zariski dense in \(X\). That would follow from a conjecture of Pink and would imply the Mordell-Lang conjecture: it would also follow from another conjecture of Zilber.
The proof is complicated. It depends on bounding the height of \(\psi(p)\) for \(p\in X^{{\text{oa,}}[s]}\) and suitable \(\psi: A\to B\), and doing so in a rather precise way. Other essential ingredients are a result of Ax bounding the dimensions of the subgroups of \(A\) generated by certain subsets, and generalities about abelian varieties such as the theorem of the cube. Because the latter uses completeness in an essential way, a proof for semiabelian varieties would have to be significantly different.

MSC:
14K15 Arithmetic ground fields for abelian varieties
11G50 Heights
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties
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