  In this paper, we introduce a sub-family of the usual generalized Wronskians,
that we call geometric generalized Wronskians. It is well-known that one can
test linear dependance of holomorphic functions (of several variables) via the
identical vanishing of generalized Wronskians. We show that such a statement
remains valid if one tests the identical vanishing only on geometric
generalized Wronskians. It turns out that geometric generalized Wronskians
allow to define intrinsic objects on projective varieties polarized with an
ample line bundle: in this setting, the lack of existence of global functions
is compensated by global sections of powers of the fixed ample line bundle.
Geometric generalized Wronskians are precisely defined so that their local
evaluations on such global sections globalize up to a positive twist by the
ample line bundle. We then give three applications of the construction of
geometric generalized Wronskians: one in intermediate hyperbolicity, and two in
foliation theory. In intermediate hyperbolicity, we show the algebraic
degeneracy of holomorphic maps from C p to a Fermat hypersurface in P N of
degree $\delta$ > (N + 1)(N -- p): this interpolates between two well-known
results, namely for p = 1 (first proved via Nevanlinna theory) and p = N -- 1
(in which case the Fermat hypersurface is of general type). The first
application in foliation theory provides a criterion for algebraic
integrability of leaves of foliations: our criterion is not optimal in view of
current knowledges, but has the advantage of having an elementary proof. Our
second application deals with positivity properties of adjoint line bundles of
the form K F + L, where K F is the canonical bundle of a regular foliation F on
a smooth projective variety X, and where L is an ample line bundle on X.
