Title: The complex case of Schmidt's going-down Theorem,
Abstract: In 1967, Schmidt wrote a seminal paper [10] on heights of subspaces of R n or
C n defined over a number field K, and diophantine approximation problems. The
going-down Theorem -- one of the main theorems he proved in his paper --
remains valid in two cases depending on whether the embedding of K in the
complex field C is a real or a complex non-real embedding. For the latter, and
more generally as soon as K is not totally real, at some point of the proof,
the arguments in [10] do not exactly work as announced. In this note, Schmidt's
ideas are worked out in details and his proof of the complex case is presented,
solving the aforementioned problem. Some definitions of Schmidt are
reformulated in terms of multilinear algebra and wedge product, following the
approaches of Laurent [5], Bugeaud and Laurent [1] and Roy [7], [8]. In [5]
Laurent introduces in the case K = Q a family of exponents and he gives a
series of inequalities relating them. In Section 5 these exponents are defined
for an arbitrary number field K. Using the going-up and the going-down Theorems
Laurent's inequalities are generalized to this setting.