Winter School on New Applications of Mixed Hodge Modules: January 15-26, 2024

Dougal Davis (University of Melbourne)

Title: Mixed Hodge modules and representation theory

Abstract: Mixed Hodge modules have a long history of applications in representation theory, going right back to their inception in the 1980s. The main goal of these lectures is to discuss some very recent work in this tradition by myself and Kari Vilonen, based on an older proposal of Schmid and Vilonen. The focus will be a version of Beilinson-Bernstein localization for mixed Hodge modules (from our preprint arXiv:2309.13215), which gives a remarkably tight relationship between mixed Hodge modules on a flag variety and Hodge-like structures on representations of complex reductive Lie algebras. I will also explain our main application: a new Hodge-theoretic characterization of unitary representations of real reductive Lie groups. These theorems and their proofs will take us on a tour of both representation theory and mixed Hodge modules, and will make contact with many links between these two beautiful subjects, both old and new.

Robert Friedman (Columbia University) and Radu Laza (Stony Brook University)

Title: Higher rational singularities and deformation theory

Abstract: It has long been known that there are deep connections between the study of deformations of singularities, both local and global, and their Hodge theory. In this lecture series, we begin with a rapid introduction to deformation theory, starting with the definition of the local and global tangent spaces to the deformation functor and the $T^1$-lifting property. Next we describe classical results on deformations of singular del Pezzo and $K3$ surfaces with rational double points or elliptic singularities. Turning to higher dimensions, we study the local theory of deformations of isolated singularities with canonical or log canonical singularities. The main new ingredient is the notion of $k$-Du Bois, $k$-rational, and $k$-liminal singularities, especially for $k=0,1$. After defining these singularities, we study their basic properties, especially how they imply local vanishing theorems and local freeness of the higher direct images of relative K\”ahler differentials in flat proper families. Finally, we apply these results to the study of the deformation theory of Fano and Calabi-Yau varieties with isolated hypersurface singularities in dimension at least three.

Christian Schnell (Stony Brook University)

Title: Hodge theory and Lagrangian fibrations

Abstract: I am going to talk about my paper (arXiv:2303.05364) on Lagrangian fibrations from earlier this year. In a nutshell, you take a proper Lagrangian fibration (on a holomorphic symplectic manifold that is Kähler but not necessarily compact, such as the Hitchin fibration on the moduli space of Higgs bundles), and then look at the Hodge modules on the base that you get from the decomposition theorem. There are several very unexpected symmetries among the graded pieces of their de Rham complexes, and I want to explain how these come about. Along the way, I plan to review two elements of the general theory: (1) how to compute direct images along proper morphisms using smooth differential forms; (2) the BGG correspondence, which allows one to reconstruct the associated graded of a filtered D-module (M,F) from the graded pieces of its de Rham complex.

Mihnea Popa (Harvard University)

Title:  The Du Bois complex

Abstract: My two lectures are meant to provide a down to earth introduction to the Du Bois complex of a singular variety, and to some of its applications. In the first lecture I will introduce the definition of Du Bois complexes via hyper-resolutions, describe some of its main properties, and compute some examples. In the second I will discuss further topics, like the interpretation in terms of mixed Hodge modules, or Du Bois singularities. New developments involving more subtle aspects of the Du Bois complex, including refinements of the notions of Du Bois and rational singularities, will be touched upon briefly at the end.

Qianyu Chen (University of Michigan) and Mircea Mustata (University of Michigan)

Title: Minimal exponent and singularities

Abstract: The goal of these lectures is to give an introduction to the minimal exponent, an invariant of hypersurface singularities that refines the log canonical threshold. We will begin with the definition in terms of Bernstein-Sato polynomials, discuss the characterization via the V-filtration with respect to the graph embedding and via the Hodge/pole order filtrations on local cohomology, and use these results to prove the basic properties of this invariant. We will then describe the role of the minimal exponent in characterizing two classes of singularities: the higher Du Bois and higher rational singularities. The key ingredient in this study is the theory of Hodge modules. If there will be time, at the end, we might discuss an extension of this circle of ideas to local complete intersections.

Ben Davison (University of Edinburgh)

Title: Purity and positivity in Donaldson-Thomas theory

Abstract: Donaldson-Thomas theory is a branch of enumerative geometry, which assigns numbers to certain moduli spaces of objects in 3-Calabi-Yau (3CY) categories: examples include the category of coherent sheaves on a Calabi-Yau threefold, local systems on a real three-manifold, and representations of Jacobi algebras. There is an enriched version of the theory in which these numbers are recovered as dimensions of vector spaces, and an even more enriched version of the theory in which these vector spaces carry mixed Hodge structures, and are themselves obtained by taking the derived global sections of certain mixed Hodge modules. This series of lectures will cover the use of MHMs to prove structural results in cohomological DT theory, like the integrality conjecture. This result states that the vanishing cycle cohomology of stacks of objects in 3CY categories are built in a precise way out of a more manageable mixed Hodge structure called BPS cohomology. The main goal of the first half of the lecture series will be to state and prove this conjecture, using Saito’s version of the decomposition theorem and some Hall algebra ideas. Then we will move on to applications of the use of MHMs in DT theory. We will see how the purity of certain MHMs and mixed Hodge structures can be used to (re)prove Kac’s positivity conjecture, extend the BBDG decomposition theorem, and provide an extension of the nonabelian Hodge isomorphism to the case of moduli stacks.

Robert Friedman (Columbia University) and Radu Laza (Stony Brook University)

Title: Higher rational singularities and deformation theory

Abstract: It has long been known that there are deep connections between the study of deformations of singularities, both local and global, and their Hodge theory. In this lecture series, we begin with a rapid introduction to deformation theory, starting with the definition of the local and global tangent spaces to the deformation functor and the $T^1$-lifting property. Next we describe classical results on deformations of singular del Pezzo and $K3$ surfaces with rational double points or elliptic singularities. Turning to higher dimensions, we study the local theory of deformations of isolated singularities with canonical or log canonical singularities. The main new ingredient is the notion of $k$-Du Bois, $k$-rational, and $k$-liminal singularities, especially for $k=0,1$. After defining these singularities, we study their basic properties, especially how they imply local vanishing theorems and local freeness of the higher direct images of relative K\”ahler differentials in flat proper families. Finally, we apply these results to the study of the deformation theory of Fano and Calabi-Yau varieties with isolated hypersurface singularities in dimension at least three.