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Mathematics Colloquium

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Date:  Tuesday, April 17, 2018

Title:  Hodge Theory and o-minimal geometry

Abstract:  (joint w. Ben Bakker) Hodge theory studies algebraic varieties by studying the periods of its global differential forms. It gives a way to assign to every algebraic variety X a linear algebraic object called a "Hodge Structure", and the famous Hodge conjecture states that one can understand much about the geometry of X by studying the associated hodge structures. One fruitful way of understanding hodge structures is by looking at their moduli space M, which can naturally be given the structure of a complex orbifold. In the case of weight 1 structures, M parametrizes abelian varieties and so is naturally an algebraic variety. However, in the general case it is known that M does not admit an algebraic structure. This creates a difficult situation, since families of algebraic varieties over an algebraic base B give holomorphic maps (known as period mappings) B-->M, but holomorphic maps can behave very badly in general (for instance, their asymptotics can be quite unwieldy, as opposed to algebraic maps). We explain how to provide a partial substitute for the lack of an algebraic structure by equipping M with an o-minimal structure, and show that the period mappings are "definable" with respect to this structure. It turns out that o-minimality gives an extremely useful notion of "tameness"; for instance, a very powerful theorem of Peterzil-Starchenko says that holomorphic maps which are o-minimal have to be algebraic in a wide variety of circumstances. As a consequence of this work, we give an easy proof of a result of Cattani-Deligne-Kaplan giving evidence towards the Hodge conjecture. The proof of our main theorem relies heavily on work of Kashiwara, Schmid and Cattani-Kaplan-Schmid on asymptotics of Period mappings.


Speaker:  Jacob Tsimerman
Institution:  University of Toronto

 

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