On the Hodge conjecture for hypersurfaces in toric varieties

TL;DR

This paper proves that very general hypersurfaces in certain toric varieties satisfy the Hodge conjecture, establishing a link between the Oda and Hodge conjectures, with explicit combinatorial criteria provided.

Contribution

It demonstrates the Hodge conjecture for a class of hypersurfaces in toric varieties and connects it to the Oda conjecture with explicit degree-based criteria.

Findings

Hodge conjecture holds for very general hypersurfaces in specific toric varieties

Establishes a connection between the Oda and Hodge conjectures

Provides explicit combinatorial criteria depending on degree

Abstract

We show that very general hypersurfaces in odd-dimensional simplicial projective toric varieties verifying a certain combinatorial property satisfy the Hodge conjecture (these include projective spaces). This gives a connection between the Oda conjecture and Hodge conjecture. We also give an explicit criterion which depends on the degree for very general hypersurfaces for the combinatorial condition to be verified.