Variational Hodge conjecture and Hodge loci

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• Hossein Movasati (IMPA - Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro)
• Thursday 23 January 2020, 11:15-12:15
• Seminar Room 2, Newton Institute.

If you have a question about this talk, please contact INI IT.

KAH - K-theory, algebraic cycles and motivic homotopy theory

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.

This talk is part of the Isaac Newton Institute Seminar Series series.

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