We prove that the algebraic multiplicity of a holomorphic vector field at an isolated singularity is invariant by $C^1$ equivalences.

We confirm a conjecture of Bernstein–Lunts which predicts that the characteristic variety of a generic polynomial vector field has no homogeneous involutive subvarieties besides the zero section and subvarieties of fibers over singular points.

Let $X$ be a normal projective variety, and let $A$ be an ample Cartier divisor on $X$. Suppose that $X$ is not the projective space. We prove that the twisted cotangent sheaf ${\Omega }_X\otimes A$ is generically nef with respect to the polarisation $A$. As an application we prove a Kobayashi-Ochiai theorem for foliations: if $ℱ⊊T_X$ is a foliation such that $detℱ\equiv i_{ℱ}A$, then $i_{ℱ}$ is at most the rank of $ℱ$.