Logarithmic vector fields and foliations on toric varieties

TL;DR

This paper develops a toric analogue of the sheaf of logarithmic vector fields, generalizing existing concepts and establishing connections with foliations and algebraic independence in the Cox ring.

Contribution

It introduces a new toric version of the logarithmic vector field sheaf, generalizes the Saito criterion for freeness, and explores its relation to foliations on toric varieties.

Findings

Established a toric sheaf of logarithmic vector fields

Generalized Saito's criterion for freeness in the toric setting

Linked the sheaf to holomorphic foliations on toric varieties

Abstract

We introduce a toric version of the sheaf of logarithmic vector fields along a divisor of a simplicial toric variety. The notion is also relevant for algebraically independent families of polynomials in the Cox ring. We provide a generalisation of the Saito criterion for the freeness of the toric logarithmic sheaf. We explain the relationship between this sheaf and the usual sheaf of logarithmic vector fields and the connection with holomorphic foliations on toric varieties.