Hodge theory and compact moduli of algebraic varieties

For any complex algebraic variety X, the singular cohomology of X is an important topological invariant.  This invariant can be enhanced to a Hodge structure, which additionally records the integrals of algebraic differential forms along topological cycles, and therefore also sees the algebraic structure.  Hodge structures turn out to be extremely powerful tools in many areas of mathematics, for example in moduli theory:  any family of algebraic varieties yields a family (or "variation") of Hodge structures, which can often be used to build a moduli space of those varieties.  The downside is that since Hodge structures are by their very nature highly transcendental, it is more difficult to realize these moduli spaces as algebraic varieties themselves.

Famously, this story plays out very elegantly for the moduli space A_g of (principally polarized) g-dimensional abelian varieties where this perspective gives an exact moduli space, and the work of Baily--Borel provides a beautiful projective compactification of this space which connects the Hodge theory to the classical theory of automorphic forms.

In this lecture series, I will explain some recent joint work with S. Filipazzi, M. Mauri, and J. Tsimerman which generalizes this picture to any family by constructing Baily--Borel compactifications for arbitrary variations of Hodge structures.  This will have especially nice applications to the moduli of Calabi--Yau varieties, and in particular implies the b-semiampleness conjecture.  The rough plan will be as follows: 

Lecture 1: A quick tour of o-minimality and o-minimal GAGA.

Lecture 2: Semiampleness of Hodge bundles.

Lecture 3: Baily--Borel compactifications, b-semiampleness, and compact moduli of Calabi--Yau varieties.

References

1. Hodge theory and o-minimality, https://benjamin-bakker.github.io/bonn.current.pdf.

2. o-minimal GAGA and a conjecture of Griffiths, with Y. Brunebarbe and J. Tsimerman, Invent. Math. 232(1):163-228 (2023).

3. Baily–Borel compactifications of period images and the b-semiampleness conjecture, with S. Filipazzi, M. Mauri, and J. Tsimerman, https://arxiv.org/abs/2508.19215.   

An introduction to wall-crossing for moduli of varieties 

In this mini-course, we will introduce moduli of Fano varieties and pairs and moduli of canonically polarized varieties and pairs (X,D).  By varying the coefficient of the divisor D, these moduli spaces exhibit a wall-crossing phenomenon.  We will explain the notion of wall-crossing and briefly outline the ideas of the proofs of the wall-crossing theorems.  We will then apply this to several particular problems, including some explicit moduli problems and moduli of Calabi Yau varieties and pairs. 

Motivic invariants of birational maps

These lectures are based on joint work with Evgeny Shinder.

Lecture 1 : Birational maps and their motivic invariants

We start by some problems about birational automorphism groups, which serve as one of the motivations for introducing motivic invariants. We then construct motivic invariants c(f) of birational maps f : X - -> Y, which are additive invariants measuring the difference between the birational types of the exceptional divisors of f and those of its inverse f^{−1}. We explain how these invariants can be used to obtain solutions to some of these problems, based on non-vanishing results presented in Lecture 2.

Lecture 2 : Vanishing and non-vanishing results

We show that the motivic invariants vanish on the birational automorphism group Bir(X) when X is a surface over a perfect field or a complex threefold. On the other hand, we construct some birational transformations f of P^N such that c(f) is nonzero. We finish by presenting some open problems and conjectures.

Lecture 3 : On the unboundedness of birational automorphisms 

We formulate boundedness questions for birational automorphisms and discuss to which extent the Cremona groups exhibit unbounded behavior. As an example, based on motivic invariants we show that over a field of characteristic zero, the weak factorization centers occurring in the birational transformations of P^N do not belong to a birationally bounded family whenever N ≥ 4, even after cancellations up to birational equivalence of their MRC bases.

References:

1. Motivic invariants of birational maps, Annals of Math. 199 (2024), No. 1, 445-478. 

2. Unboundedness for motivic invariants of birational automorphisms, arXiv:2510.00290. 

Lectures on K-stability

We will discuss the recent progress on K-stability theory of Fano varieties, including different characterizations of K-stability, the construction of K-moduli spaces. If time permits, we will also survey the progress on the local stability theory for klt singularities.