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Differential equations and mixed Hodge structures

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KA2W02 - Arithmetic geometry, cycles, Hodge theory, regulators, periods and heights

We report on a new development in asymptotic Hodge theory, arising from work of Golyshev-Zagier and Bloch-Vlasenko, and connected to the Gamma Conjectures in Fano/LG-model mirror symmetry.  The talk will focus exclusively on the Hodge/period-theoretic aspects.   Given a variation of Hodge structure M on a Zariski open in P^1, the periods of the limiting mixed Hodge structures at the punctures are interesting invariants of M.  More generally, one can try to compute these asymptotic invariants for iterated extensions of M by “Tate objects”, which may arise for example from normal functions associated to algebraic cycles.     The main point of the talk will be that (with suitable assumptions on M) these invariants are encoded in an entire function called the motivic Gamma function, which is determined by the Picard-Fuchs operator L underlying M.  In particular, when L is hypergeometric, this is easy to compute and we get a closed-form answer (and a limiting motive).     Though that is probably enough for a single talk, perhaps one more thing is worth mentioning in this abstract:  in the next simplest class of cases beyond hypergeometric, the leading Taylor coefficient of the motivic Gamma at 1 is given by the special value of a normal function, and in one special case this recovers Apery’s irrationality proof for zeta(3).

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

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