Abstract

Grothendieck’s variational Hodge conjecture (VHC) claims that if we have a continuous family of Hodge cycles  (flat section of the Gauss-Manin connection) and the Hodge conjecture is true at least for one Hodge cycle of the family then it must be true for all such Hodge cycles. A stronger version of this (Alternative Hodge conjecture, AHC ),  asserts that the deformation of an algebraic cycle Z togther with the projective variety X, where it lives,  is the same as the deformation of the cohomology class of Z in X. There are many simple counterexamples to AHC , however, in explict situations, like algebraic cycles inside hypersurfaces, it becomes a challenging problem. In  this talk I will review few cases in which AHC is true (including Bloch's semi-regular and local complete intersection  algebraic cycles) and other cases in which it is not true.   The talk is mainly based on the article  arXiv:1902.00831.