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

Author(s)

Bruzzo, Ugo ^{[1, 2, 3, 4, 5]} ; Montoya, William ^[6]

Affiliation

^[1] IGAP Inst Geometry & Phys, Trieste - Italy

^[2] INFN Ist Nazl Fis Nucl, Sez Trieste, Trieste - Italy

^[3] Arnold Regge Ctr Algebra Geometry & Theoret Phys, Turin - Italy

^[4] SISSA Scuola Int Studi Avanzati, Via Bonomea 265, I-34136 Trieste - Italy

^[5] Univ Fed Paraiba, Dept Matemat, Campus 1, BR-58051900 Joao Pessoa, Paraiba - Brazil

^[6] Univ Estadual Campinas, Inst Matemat Estat & Comp Cient, Rua Sergio Buarque de Holanda 651, BR-13083859 Campinas, SP - Brazil

• Document type: Journal article
• Source: SAO PAULO JOURNAL OF MATHEMATICAL SCIENCES; JUL 2021.
• Web of Science Citations: 0

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 Bruzzo and Grassi (Commun Anal Geom 28: 1773-1786, 2020). 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, under the assumption that the ambient variety P-Sigma(2k+1) has Picard group Z. This extends to a class of toric varieties Otwinowska's result in Otwinowska (J Alg Geom 12: 307-320, 2003). (AU)

• FAPESP's process: 19/23499-7 - Noether-Lefschetz theory in toric varieties
• Grantee: William Daniel Montoya Cataño
• Support Opportunities: Scholarships in Brazil - Post-Doctoral