A Weak (k,k)-Lefschetz Theorem for Projective Toric Orbifolds

TL;DR

This paper generalizes the Lefschetz theorem for projective toric orbifolds and proves the Hodge conjecture for certain 2k-dimensional hypersurfaces derived via the Cayley trick.

Contribution

It extends the (1,1)-Lefschetz theorem to a broader class of toric orbifolds and establishes algebraicity of rational (k,k)-cohomology classes on specific hypersurfaces.

Findings

Generalized (1,1)-Lefschetz theorem for projective toric orbifolds

Proved Hodge conjecture for certain 2k-dimensional hypersurfaces

Demonstrated algebraicity of rational (k,k)-cohomology classes

Abstract

Firstly we show a generalization of the (1,1)-Lefschetz theorem for projective toric orbifolds and secondly we prove that on 2k-dimensional quasi-smooth hypersurfaces coming from quasi-smooth intersection surfaces, under the Cayley trick, every rational (k,k)-cohomology class is algebraic, i.e., the Hodge conjecture holds on them.