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SUMMARY:Variational Hodge conjecture and Hodge loci - Hossein Movasati (IM
 PA - Instituto Nacional de Matemática Pura e Aplicada\, Rio de Janeiro)
DTSTART:20200123T111500Z
DTEND:20200123T121500Z
UID:TALK137980@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:Grothendieck&rsquo\;s variational Hodge conjecture (VHC) claim
 s that if we have a continuous family of Hodge cycles&nbsp\; (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)\,&
 nbsp\; asserts that the deformation of an algebraic cycle Z togther with t
 he projective variety X\, where it lives\,&nbsp\; is the same as the defor
 mation of the cohomology class of Z in X. There are many simple counterexa
 mples to AHC\, however\, in explict situations\, like algebraic cycles ins
 ide hypersurfaces\, it becomes a challenging problem. In&nbsp\; this talk 
 I will review few cases in which AHC is true (including Bloch&#39\;s semi-
 regular and local complete intersection&nbsp\; algebraic cycles) and other
  cases in which it is not true.&nbsp\;&nbsp\; The talk is mainly based on 
 the article&nbsp\; arXiv:1902.00831. <br><br><br><br><br>
LOCATION:Seminar Room 2\, Newton Institute
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