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Period integrals of algebraic varieties
University of Chicago, November 2021.
Abstract.
A period integral of a complex algebraic variety is the integral of an algebraic differential form along a topological cycle. While such integrals are transcendental in nature, they have had pervasive importance in algebraic and arithmetic geometry going back to the foundational work of Abel and Jacobi. The functions one obtains from taking period integrals in families of algebraic varieties moreover recur in many areas of mathematics. In this talk I will first survey how algebraic information can be extracted from period integrals, as well as their connection to various conjectural frameworks. I will then discuss recent progress on understanding the algebraic properties of period integrals in families and recent applications to Hodge theory, algebraic geometry, arithmetic geometry, and logic.
- 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 Kahler 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 Kahler variety can be approximated by algebraic varieties.
- Quasiprojectivity of images of mixed period maps
Princeton, October 2020.
Abstract.
Families of smooth proper algebraic varieties give rise to variations of pure Hodge structures; general algebraic families yield variations of mixed Hodge structures. It was conjectured by Griffiths and proven in joint work with Y. Brunebarbe and J. Tsimerman that the closure of the image of the classifying map to the moduli space of Hodge structures is a quasiprojective algebraic variety in the pure case. In this talk I will explain how to extend this result to the mixed setting. As in the pure case, the proof heavily uses techniques from o-minimal geometry, and we will also discuss some related applications.
- Recent developments in the moduli of singular symplectic varieties
Categories, Cycles and Cohomology of Hyperkahler Varieties, September 2020.
Abstract.
A recurring feature of hyperkahler manifolds is that their geometry is closely tied to the Hodge structure on their weight 2 cohomology. Verbitsky's global Torelli theorem, for instance, essentially shows that for any hyperkahler manifold, all deformations of the Hodge structure are realized geometrically exactly once, at least up to bimeromorphism. In this talk I will survey joint work with C. Lehn extending these moduli-theoretic results to the case of singular symplectic varieties, including a generalization of the global Torelli theorem. We will also discuss some more recent work with C. Lehn and H. Guenancia extending the decomposition of projective symplectic varieties by holonomy type due to Druel--Greb--Guenancia--Horing--Kebekus--Peternell to the nonprojective case.
- Hodge theory and o-minimality
Brown, February 2020.
Abstract.
The cohomology groups of complex algebraic varieties come equipped with a powerful but intrinsically analytic invariant called a Hodge structure. Hodge structures of certain very special algebraic varieties are nonetheless parametrized by algebraic varieties, and while this leads to many important applications in algebraic and arithmetic geometry it fails badly in general. Recent joint work with Y. Brunebarbe, B. Klingler, and J. Tsimerman remedies this failure by showing that parameter spaces of Hodge structures always admit "tame" analytic structures in a sense made precise using ideas from model theory. A salient feature of the resulting tame analytic geometry is that it allows for the local flexibility of the full analytic category while preserving the global behavior of the algebraic category.
In this talk I will explain this perspective as well as some important applications, including an easy proof of a celebrated theorem of Cattani--Deligne--Kaplan on the algebraicity of Hodge loci and the resolution of a longstanding conjecture of Griffiths on the quasiprojectivity of the images of period maps.
University of Illinois at Chicago