
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 2form. 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.

The geometric AndréGrothendieck 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 AxSchanuel 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 AndreGrothendieck period conjecture, which predicts that all
algebraic relations between such periods integrals arise from
geometry. This is joint work with J. Tsimerman.

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.