Abstract
AbstractWe 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 $${{\mathbb {P}}}_\Sigma ^{2k+1}$$
P
Σ
2
k
+
1
has Picard group $${\mathbb {Z}}$$
Z
. This extends to a class of toric varieties Otwinowska’s result in Otwinowska (J Alg Geom 12: 307–320, 2003).
Funder
Scuola Internazionale Superiore di Studi Avanzati - SISSA
Publisher
Springer Science and Business Media LLC
Subject
Computational Theory and Mathematics,Statistics, Probability and Uncertainty,General Mathematics
Reference18 articles.
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2. Bruzzo, U., Grassi, A.: The Noether-Lefschetz locus of surfaces in toric threefolds, Commun. Contemp. Math. No. 5 (2017) 1750070 (20 pages)
3. Bruzzo, U., Grassi, A.: On the Hodge conjecture for hypersurfaces in toric varieties. Commun. Anal. Geom. 28, 1773–1786 (2020)
4. Bruzzo, U., Montoya, W.D.: An asymptotic description of the Noether-Lefschetz components in toric varieties, arXiv:1905.01570 [math.AG]
5. Bruzzo, U., Montoya, W.D.: Codimension bounds for the Noether–Lefschetz components for toric varieties, arXiv:2001.01960 [math.AG]. To appear in Eur. J. Math
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