On the Hodge conjecture for quasi-smooth intersections in toric varieties

TL;DR

This paper proves the Hodge conjecture for certain quasi-smooth intersections in toric varieties, extending previous results to more general subvarieties and hypersurfaces with large degrees.

Contribution

It establishes the Hodge conjecture for very general quasi-smooth intersections and asymptotically for high-degree hypersurfaces in toric varieties, generalizing prior work.

Findings

Hodge conjecture holds for very general quasi-smooth intersections

Asymptotic Hodge conjecture validity for high-degree hypersurfaces

Extension of Otwinowska's results to toric varieties

Abstract

We establish the Hodge conjecture for some subvarieties of a class of toric varieties. First we study quasi-smooth intersections in a projective simplicial toric variety, which is a suitable notion to generalize smooth complete intersection subvarieties in the toric environment, and in particular quasi-smooth hypersurfaces. We show that under appropriate conditions, the Hodge Conjecture holds for a very general quasi-smooth intersection subvariety, generalizing the work on quasi-smooth hypersurfaces of the first author and Grassi in [3]. We also show that the Hodge Conjecture holds asymptotically for suitable quasi-smooth hypersurface in the Noether-Lefschetz locus, where "asymptotically" means that the degree of the hypersurface is big enough. This extendes to toric varieties Otwinowska's result in [15].