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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.
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Period integrals of algebraic varieties
Non-abelian Hodge theory, Saint-Jacut de la Mer, June 2022.
Abstract.
Period integrals on complex algebraic varieties are the integrals of algebraic differential forms along topological cycles. They are at the heart of Hodge theory. In this talk I will survey some recent results on the behavior of the functions obtained by taking period integrals in algebraic families. In particular I will discuss the proof of an Ax–Schanuel type theorem on the transcendence of these functions and show it is equivalent to a geometric version of the André–Grothendieck period conjecture. This is joint work with J. Tsimerman.
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Compact hyperkähler varieties: basic results
Derived Categories, Moduli Spaces, and Hyperkähler Varieties, Ann Arbor, August 2022.
Abstract.
Compact hyperkähler manifolds enjoy a number of nice properties, many of which are connected to the Hodge structure on their weight 2 cohomology. Surprisingly, much of this theory extends to the case of singular compact hyperkähler varieties, which arise naturally even in the study of hyperkähler manifolds and are interesting in and of themselves. The goal of these lectures is to introduce the basic objects and survey some important recent developments. Topics will include: basic definitions and examples, Hodge theory and deformation theory, birational geometry and the global Torelli theorem, and the Beauville-Bogomolov decomposition theorem.
- Algebraic approximation of Calabi–Yau varieties and the decomposition theorem
Geometry and TACoS, November 2020.
Abstract.
Calabi–Yau manifolds are built out of simple pieces by the Beauville–Bogomolov decomposition theorem: any Calabi–Yau Kähler manifold up to an etale cover is a product of complex tori, irreducible holomorphic symplectic manifolds, and strict Calabi–Yau manifolds (which have no holomorphic forms except a holomorphic volume form). Work of Druel–Guenancia–Greb–Horing–Kebekus–Peternell over the last decade has culminated in a generalization of this result to projective Calabi–Yau varieties with the kinds of singularities that arise in the MMP, and the proofs critically use algebraic methods. In this talk I will describe joint work with H. Guenancia and C. Lehn extending the decomposition theorem to nonprojective varieties via deformation theory. A crucial step in the proof is the resolution of the K-trivial case of a conjecture of Peternell asserting that any minimal Kähler variety can be approximated by algebraic varieties.
University of Illinois at Chicago