Degree-one foliations on complete intersections

TL;DR

This paper classifies degree-one codimension-one foliations on certain smooth projective varieties, showing they form two main types under mild conditions, with implications for foliations on manifolds covered by lines.

Contribution

It establishes the irreducible components of the space of such foliations on complete intersections and certain hypersurfaces, extending understanding of their structure.

Findings

Two irreducible components of logarithmic type for foliations on complete intersections.

Similar classification applies to hypersurfaces of dimension at least three, excluding quadrics.

Results follow from a general structure theorem for foliations on line-covered manifolds.

Abstract

We prove that, under mild restrictions, the space of codimension-one foliations of degree one on a smooth projective complete intersection has two irreducible components of logarithmic type. We also prove that the same conclusion holds for any smooth hypersurface of dimension at least three that is not a quadric threefold. The proof of these results follows essentially from a more general structure theorem for foliations on manifolds covered by lines.