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arXiv · 2102.01050

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

Ugo Bruzzo·William D. Montoya
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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].

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Ugo Bruzzo, William D. Montoya. 2021-02-01. On the Hodge conjecture for quasi-smooth intersections in toric varieties. https://arxiv.org/abs/2102.01050

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