On the Poincar\'e problem for foliations on compact toric orbifolds

TL;DR

This paper establishes an optimal upper bound on the degree of quasi-smooth hypersurfaces invariant under holomorphic foliations on compact toric orbifolds, linking the bound to the foliation's degree and toric coordinate degrees.

Contribution

It provides the first sharp bound for invariant hypersurfaces on compact toric orbifolds, advancing understanding of foliation invariants in this geometric setting.

Findings

Derived an explicit upper bound depending on foliation and coordinate degrees

Bound is proven to be optimal for the class of hypersurfaces considered

Results apply to complete simplicial toric varieties

Abstract

We give an optimal upper bound of the degree of quasi-smooth hypersurfaces which are invariant by a one-dimensional holomorphic foliation on a compact toric orbifold, i.e. on a complete simplicial toric variety. This bound depends only on the degree of the foliation and of the degrees of the toric homogeneous coordinates.