Abstract: I'll explain two opposing pieces of work: (1) Markman's proof of the Hodge conjecture for general Weil type abelian fourfolds of discriminant 1, and (2) Kontsevich's tropical approach to finding a counterexample to the Hodge conjecture for Weil type abelian varieties. Then I'll explain why Markman's proof of the Hodge conjecture in the discriminant 1 case rules out Kontsevich's approach in dimension 4 (for arbitrary discriminant). This last observation is pretty elementary, but I think it illustrates some of the techniques that go into working with Mumford-Tate groups in a nice way. The observation itself is part of joint work in progress that I'm doing with Helge Ruddat.
Abstract: Given a system of linear differential equations on a space X, one gets a representation of the fundamental group of X by considering the monodromy of the system. The Riemann-Hilbert problem asks the converse: what systems of linear differential equations arise with prescribed monodromy representations? In this talk, I will discuss Deligne's solution to the Riemann-Hilbert problem for smooth, connected quasi-projective varieties over the complex numbers, following a survey by Nicholas Katz: https://web.math.princeton.edu/~nmk/old/DeligneXXIHilbert.pdf
Abstract: Given an analytic family of smooth projective varieties over a complex manifold B, one can construct a holomorphic vector bundle over B whose fibers carry a polarized Hodge structure of weight k. This family of Hodge structures can be abstracted to the notion of a variation of Hodge structure (VHS) over B. In this talk, I will first describe the geometric VHS in the above setting and then define the abstract notion of VHS following E. Cattani's notes on VHS: https://webusers.imj-prg.fr/~fouad.elzein/Hodge.pdf
Abstract: Given an analytic family of smooth projective varieties over a complex manifold B, one can construct a holomorphic vector bundle over B whose fibers carry a polarized Hodge structure of weight k. This family of Hodge structures can be abstracted to the notion of a variation of Hodge structure (VHS) over B. In this talk, I will first describe the geometric VHS in the above setting and then define the abstract notion of VHS following E. Cattani's notes on VHS: https://webusers.imj-prg.fr/~fouad.elzein/Hodge.pdf
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