Geometric Generalized Wronskians: Applications in Intermediate Hyperbolicity and Foliation Theory

Abstract

Abstract In this paper, we introduce a sub-family of the usual generalized Wronskians, which 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 ${\mathbb {C}}^{p}$ to a Fermat hypersurface in $\textbf {P}^{N}$ 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 knowledge, but has the advantage of having an elementary proof. Our second application deals with positivity properties of adjoint line bundles of the form $K_{\mathcal {F}} + L $, where $K_{\mathcal {F}}$ is the canonical bundle of a regular foliation $\mathcal {F}$ on a smooth projective variety $X$, and where $L$ is an ample line bundle on $X$.