Multiplicative equations on torus subvarieties

作者：Maurin G^*

来源：International Mathematics Research Notices, 2011, (23): 5259-5366.

DOI：10.1093/imrn/rnq248

Given an algebraic subvariety X of a multiplicative torus A over , we study the intersection of with the union A([m]) of all algebraic subgroups of codimension . This is related to conjectures stated by Zilber and Pink and follows previous work by Bombieri, Masser and Zannier. Our main result can be stated as follows: if X is a nondegenerate subvariety of A, then for any subgroup of finite rank Gamma of , the points lying in a height-theoretic cone with nonzero radius around Gamma A([m]) are not dense in X. The case Gamma={1} was proved by Habegger in 2009 as a corollary to his bounded height theorem. We show here that the general case follows from a new Vojta-type height inequality on X(m). The proof of this inequality relies on diophantine approximation techniques coming from previous work by Vojta, Faltings, Bombieri, ... , Remond. It shows that the height inequality can be deduced from a lower bound on some intersection numbers coming from a family of blow-ups of a given compactification of X(m). Our main task is to establish this lower bound. For that purpose, we use homogeneity and positivity properties of the intersection numbers. They are proved here in an analytic setting, using integrals of Chern-like forms over suitably chosen desingularizations.

• 出版日期2011