Quantitative Diophantine Approximations on Projective Varieties

In this note we study some aspects of a paper of G. Faltings and G. Wüstholz [7]. In particular, we work out a quantitative version of their main theorem. Our result even gives an explicit presentation of the structure for the exceptional subspaces of [24]. For a generic projective variety we obtain double exponential estimates. However, knowing more about the geometry of the given variety one can substantially improve these bounds (for example [5], Theorem). Even it is difficult to guess what should be the optimal results (see [23]). In any case, the approach of [7] leads to non-trivial results that cannot be reached in full generality by the usual tools of diophantine approximations. Let us formulate our main result (Theorem 7.4): Let K ⊆ L be two number fields, S a finite set of places of L containing all infinite places, and d > 0 an integer. Let X ⊆ P be a subvariety defined over K. For v ∈ S, we fix non-negative reals cv,α and linear independent forms fv,α ∈ H(PL,O(d)), α = 0, · · · , ( n+d n ) . In §1 we will define a function μ associated to these datas (the slope of a family of weighted filtrations). Theorem.There exist explicitly computable positive reals c1, · · · , c5 depending on K,L, S,X, {fv,α} such that following holds: There exists a subvariety Y of X⊗K L of degree not exceeding c1 and height at most c2, such that if μ > d then all but finitely many points x ∈ X(K) with 1 [L : Q] log( |fv,α(x)|v ‖x‖v ) 6 −cv,α h(x) v ∈ S, α = 0, · · · , ( n+ d