  The aim of this paper is to establish some new results on the absolute
continuity and the convergence in total variation for a sequence of
d-dimensional vectors whose components belong to a finite sum of Wiener
chaoses. First we show that the probability that the determinant of the
Malliavin matrix of such vectors vanishes is zero or one, and this probability
equals to one is equivalent to say that the vector takes values in the set of
zeros of a polynomial. We provide a bound for the degree of this annihilating
polynomial improving a result by Kusuoka. On the other hand, we show that the
convergence in law implies the convergence in total variation, extending to the
multivariate case a recent result by Nourdin and Poly. This follows from an
inequality relating the total variation distance with the Fortet-Mourier
distance. Finally, applications to some particular cases are discussed.
