On unlikely intersections of complex varieties with tori. (English) Zbl 1162.11031

The authors consider various conjectures investigating the intersection of an algebraic variety \(\mathcal X\) of dimension \(r\) and codimension \(s\) in \(\mathbb G_m^{r+s}\) with the union \(\mathcal H_{s-1}\) of all algebraic subgroups of dimension \(s-1\). In the paper and the appendix, they list 6 conjectures which would imply that \(\mathcal X\cap\mathcal H_{s-1}\) is a comparatively small set. They already investigated this question: E. Bombieri, D. W. Masser, U. Zannier, [Mich. Math. J. 51, No. 3, 451–466 (2003; Zbl 1048.11056), Trans. Am. Math. Soc. 358, No. 5, 2247–2257 (2006; Zbl 1161.11025), Int. Math. Res. Not. 2007, No. 19, Article ID rnm057, 33 p. (2007; Zbl 1145.11049), Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 7, No. 1, 51–80 (2008; Zbl 1150.11022)].
For integers \(R\geq 0\) and \(S\geq 0\) they say that a conjecture holds for a field \(K\) of zero characteristic on the \((R,S)\) rectangle if, for all integers \(r\), \(s\) with \(0\leq r\leq R\) and \(0\leq s\leq S\), it holds for every variety of dimension \(r\) in \(\mathbb G_m^{r+s}\) defined over \(K\) and irreducible over \(\overline{K}\). For each of the above mentioned 6 conjectures, they prove that if it holds for \(\overline{\mathbb Q}\) on the \((R,S)\) rectangle, then it holds for \(\mathbb C\) on the \((R,S)\) rectangle.


11G35 Varieties over global fields
11G50 Heights
14G25 Global ground fields in algebraic geometry
14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties
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