

Seminar of the Department of Algebra and of the Department of Algebraic Geometry (Shafarevich Seminar)
August 1, 2017 15:00, Moscow, Steklov Mathematical Institute, room 540 (Gubkina 8)






Normal functions over locally symmetric varieties
M. Kerr^{} 
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Abstract:
An algebraic cycle homologous to
zero on a variety leads to an extension of Hodgetheoretic data. In
a variational context, the resulting section of a bundle of complex
tori is called a normal function, and is used to study cycles modulo
rational or algebraic equivalence.
The archetype for interesting normal functions arises from the
Ceresa cycle, consisting of the difference of two copies of a curve
in its Jacobian. The profound geometric consequences of its
existence are evidenced in work of Nori, Hain and (most recently)
Totaro. In contrast, a theorem of Green and Voisin demonstrates the
absence of normal functions arising from cycles on very general
projective hypersurfaces of large enough degree.
Inspired by recent work of FriedmanLaza on Hermitian variation of
Hodge structure and Oort's conjecture on special subvarieties in the
Torelli locus, R. Keast and I wondered about the existence of normal
functions over étale neighborhoods of Shimura varieties. In this
talk I will explain our classification of the cases where a
GreenVoisin analogue does not hold, and where one expects
interesting cycles (and generalized cycles) to occur. I will also
give evidence that these predictions might be "sharp", and draw some
geometric consequences.
Language: English

