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Hwang's theorem on the base of a Lagrangian fibration revisited
Geometry of Hyperkähler varieties, Hangzhou, September 2022.
Abstract.
Irreducible hyperkahler manifolds are higher dimensional analogs of K3
surfaces; their geometry is tightly controlled by the existence of a
nowhere degenerate holomorphic 2-form. The only nontrivial fibration
structure f:X -> B a hyperkahler manifold X admits is a fibration by
Lagrangian tori, and for such Lagrangian fibrations the base B is
conjectured to always be isomorphic to projective space. In 2008
Hwang proved that this is the case if B is assumed to be smooth by
using the theory of varieties of minimal rational tangents on Fano
manifolds. In this talk I will present a simpler proof of this result
which leans more heavily on Hodge theory. Specifically, the main
input is a basic functoriality result coming from Hodge modules. This
is joint work with C. Schnell.
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The geometric André--Grothendieck period conjecture
Midrasha Mathematicae, December 2023.
Abstract.
A period integral of a complex algebraic variety is the integral of an
algebraic differential form along a topological cycle. These numbers
are at the heart of Hodge theory. In this talk I will explain how to
prove a version of the Ax--Schanuel conjecture for these period
integrals in families, and how it provides a capstone to the advances
in the transcendence theory of period maps made over the past decade.
I will also discuss the relationship with the functional version of
the Andre--Grothendieck period conjecture, which predicts that all
algebraic relations between such periods integrals arise from
geometry. This is joint work with J. Tsimerman.
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The Matsushita alternative
Hyperkähler varieties and related topics, September 2022.
Abstract.
Matsushita conjectured that a Lagrangian fibration of an irreducible hyperkähler manifold is either isotrivial or of maximal variation. In this talk I will show how to prove this conjecture by adapting previous work of Voisin and van Geemen. I will also deduce some applications to the density of torsion points of sections of Lagrangian fibrations.