Periods of linear algebraic cycles

Abstract

In this article we use a theorem of Carlson and Griffiths and compute periods of linear algebraic cycles inside the Fermat variety of even dimension $n$ and degree $d$. As an application, for examples of $n$ and $d$ we prove that the locus of hypersurfaces containing two linear cycles whose intersection is of low dimension, is a reduced component of the Hodge locus in the underlying parameter space. We also check the same statement for hypersurfaces containing a complete intersection algebraic cycle. Our result confirms the Hodge conjecture for Hodge cycles obtained by the monodromy of the homology class of such algebraic cycles. This is known as the variational Hodge conjecture.

10 pages

References in corpus (2)

• Noether-Lefschetz locus and a special case of the variational Hodge conjecture
• Semi-regular varieties and variational Hodge conjecture

Cited by in corpus (6)

• Periods of Complete Intersection Algebraic Cycles
• Integral Hodge conjecture for Fermat varieties
• Small codimension components of the Hodge locus containing the Fermat variety
• Variational Hodge conjecture for complete intersections on hypersurfaces in projective space
• Why should one compute periods of algebraic cycles?
• Hodge cycles for cubic hypersurfaces