Summer School
Recent perspectives on Hodge theory
September 19-23, 2022
The school will consist of four lecture series of five hours each. The short programs is below:
Javier Fresán
Periods, motivic Galois groups, and the Kontsevich-Zagier conjecture
Abstract: Periods are complex numbers that can be represented as integrals of algebraic differential forms along topological cycles on algebraic varieties. According to conjectures of Grothendieck and Kontsevich-Zagier, all polynomial relations between these numbers should have "geometric origin". This would in particular give rise to a Galois theory of periods generalising the usual Galois action on algebraic numbers. I will explain how our progress in the understanding of motives over the last twenty years has allowed for two equivalent, albeit very different unconditional constructions of the motivic Galois group. Among their striking applications, I will sketch the main ideas in Ayoub's proof of a relative version of the Kontsevich-Zagier conjecture, where periods are roughly speaking replaced by series of periods. Hodge theory enters the stage through the theorem of the fixed part.
Bruno Klingler
Hodge theory, between algebraicity and transcendence
Abstract: Hodge theory, as developed by Deligne and Griffiths, is an essential tool for analyzing the geometry and arithmetic of complex algebraic varieties. It is a crucial fact that at heart, Hodge theory is NOT algebraic. On the other hand, according to both the Hodge conjecture and the Grothendieck period conjecture, this transcendence is severely constrained. Tame geometry, whose idea was introduced by Grothendieck in the 80s, and developed by model theorists under the name "o-minimal geometry", seems a natural setting for understanding these constraints. In these lectures I will present a number of recent applications of tame geometry to several problems related to Hodge theory and periods.
Jochen Heinloth
Moduli spaces of Higgs bundles
Abstract: Moduli spaces of Higgs bundles appear in Simpson's non-abelian Hodge correspondence as an algebraic interpretation of a first cohomology group that admits an action C*, an analog of a Hodge structure. These spaces have a very rich geometry, in particular, the underlying space of complex points has several different structures as points of an algebraic variety. The interplay of these different geometries has given rise to many results and open conjectures. In the lectures we will try to explain some of these.
Geordie Williamson
Hodge theory just beyond geometry
Abstract: Classical Hodge theory is a crucial part of a parcel of remarkable statements (Hodge decomposition, the weak and hard Lefschetz theorem, the Hodge-Riemann relations) concerning the cohomology of projective complex algebraic varieties (or more generally Kähler manifolds). Over the last decades, it has been discovered that there are several settings where structures similar to the cohomology groups of classical Hodge theory show up, but where the underlying variety is "missing". Typically, the existence of Hodge like structures on such spaces provides deep results in combinatorics. Examples include the bounds on the face numbers of polytopes, and the positivity of Kazhdan-Lusztig polynomials coming from the theory of Soergel modules. Another example (where my knowledge is scarce) occurs in the theory of matroids. I will try to introduce this web of ideas. If I get my act together, I should be able to highlight the general principles present in the known examples.