Varieties with very little transcendental cohomology

TL;DR

This paper introduces the motivic dimension as a measure of transcendental cohomology in complex algebraic varieties, providing estimates and examples where this measure is small, and applying it to verify cases of the Hodge conjecture.

Contribution

It defines the motivic dimension and demonstrates its use in estimating transcendental cohomology, with applications to verifying the Hodge conjecture in specific classes of varieties.

Findings

Motivic dimension is zero when all cohomology is algebraic.

Provides bounds and examples where the motivic dimension is small.

Reexamines the Hodge conjecture in various geometric contexts.

Abstract

Given a complex algebraic variety X, we define a natural number called the motivic dimension which measures the amount of transcendental (co)homology of X. It is zero precisely when all the (co)homolgy is spanned by algebraic cycles. Most of this paper is concerned with giving estimates on this number, along with examples where it is small. As an application, we check or recheck the Hodge conjectue in a number of examples: uniruled fourfolds, rationally connected fivefolds, fourfolds fibred by surfaces with p_g=0, Hilbert schemes of a small number points on surfaces with p_g=0, and generic hypersurfaces.