Varieties with very little transcendental cohomology

Details Varieties with very little transcendental cohomology Varieties with very little transcendental cohomology

2007-06-17

arxiv.org/abs/0706.2506v1

Mathematics

Algebraic Geometry

13 pages

Scientific paper

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.

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