Communications in Analysis and Geometry

Volume 28 (2020)

Number 8

The Second of Two Special Issues in Honor of Karen Uhlenbeck’s 75th Birthday

Special-Issue Editors: Georgios Daskalopoulos (Brown University), Kefeng Liu, Chuu-Lian Terng (U. of Cal. Irvine), and Shing-Tung Yau

On the Hodge conjecture for hypersurfaces in toric varieties

Pages: 1773 – 1786



Ugo Bruzzo (SISSA (Scuola Internazionale Superiore di Studi Avanzati), Trieste, Italy; IGAP (Institute for Geometry and Physics), Trieste, Italy; and INFN (Istituto Nazionale di Fisica Nucleare), Torino, Italy)

Antonella Grassi (Department of Mathematics, University of Pennsylvania, Philadelphia, Penn., U.S.A.)


We show that for very general hypersurfaces in odd-dimensional simplicial projective toric varieties satisfying an effective combinatorial property the Hodge conjecture holds. 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.

The authors’ research was partially supported by PRIN “Geometria delle varietà algebriche,” GNSAGA-INdAM, and by the University of Pennsylvania Department of Mathematics Visitors Fund. Ugu Bruzzo is a member of the VBAC group.

Received 13 May 2018

Accepted 6 February 2019

Published 8 January 2021