Mini-courses

Camilla Felisetti: The P=W conjecture

The P=W conjecture, formulated in 2010 by de Cataldo, Hausel and Migliorini and recently proved in full generality by Maulik-Shen and Hausel-Mellit-Minets-Schiffmann in two independent works, is a cornerstone result in the non abelian Hodge theory of curves. The conjecture states the equality of two filtrations of very different origin on the cohomology of the moduli space of representations of the fundamental group of a curve, relating the topology of Hitchin systems to the Hodge theory of character varieties. In the course, we will first review the basic concepts of non abelian Hodge theory, introducing the two filtrations appearing in the conjecture. Later we will review the main concepts and techniques which intervene in the formulation of P=W and, time permitting, we will give a sketch of the proofs.

Margarida Melo: Compactified and tropical (universal) Jacobians

In this series of lectures, I will start by giving an overview of the theory of compactified Jacobians for curves with mild singularities, illustrating how the different approaches relate one to the other and the main properties of such objects. I will then consider universal compactified Jacobians and discuss different constructions and applications of such objects to the study of other moduli spaces. In the second part of the course, we will introduce tropical (universal) Jacobians and explain some of their properties. Finally, we will discuss in detail the relation between the geometry of the boundary complex of universal algebraic Jacobians with their tropical counterparts.

Isabel Vogt: Applications of vector bundles to moduli of curves

In this mini-course we will study the geometry of curves and their moduli spaces by considering maps from the curve to projective space. Deformation theory relates the local structure of relevant moduli spaces to natural vector bundles on curves. Two key players in this story are the restricted tangent bundle and the normal bundle of a curve in projective space. We will use embedded degeneration to reducible nodal curves to understand these bundles.

Abstracts

Chiara Damiolini: Parahoric Bruhat-Tits groups from Galois covers

During this week we will see how the study of moduli spaces of vector bundles over an algebraic curve X has a central focus in algebraic geometry, with numerous avenues of research also in adjacent fields. A natural generalization can be obtained by replacing vector bundles with G-torsors, for G an algebraic group. In today's talk I will focus on the case when G takes the form of a parahoric Bruhat-Tits group. These are groups defined over X which are generically reductive, and that display specific "parahoric" behaviors at finitely many points on X. In particular I will discuss how, under appropriate conditions, these groups arise from decorated principal bundles on Galois coverings of X. This description was first developed for generically split parahoric Bruhat-Tits groups by Balaji and Seshadri. I will present an extension of their result to a much more general class of groups. This is based on joint work with J. Hong.

Svetlana Makarova: Moduli problems and abelian categories

I will briefly remind why viewing moduli problems as functors allows one to recover the structure of a variety on the set of isomorphism classes of objects, and then I will talk about modern methods of studying moduli problems. The modern theory ​"Beyond GIT" provides a "coordinate-free" way of thinking about classification problems. Among giving a uniform philosophy, this allows to treat problems that can't necessarily be described as global quotients, like objects in an abstract abelian category. I will show how the methods of BGIT can be applied to prove existence and projectivity of moduli spaces of objects in a class of abelian categories. This is based on a joint work with Andres Fernandez Herrero, Emma Lennen. Further, applying these methods to moduli of quiver representations allows us to focus on geometry and obtain new results. I will define a determinantal line bundle which descends to a semiample line bundle on the moduli space and provide effective bounds for its global generation. For an acyclic quiver, we can observe that this line bundle is ample and thus the adequate moduli space is projective over an arbitrary noetherian base. This part is based on a preprint with Belmans, Damiolini, Franzen, Hoskins, Tajakka (https://arxiv.org/abs/2210.00033).

Laura Pertusi: Non-commutative abelian surfaces and generalized Kummer varieties.

Examples of non-commutative K3 surfaces arise from semiorthogonal decompositions of the bounded derived category of certain Fano varieties. The most interesting cases are those of cubic fourfolds and Gushel-Mukai varieties of even dimension. Using the deep theory of families of stability conditions, locally complete families of hyperkahler manifolds deformation equivalent to Hilbert schemes of points on a K3 surface have been constructed from moduli spaces of stable objects in these non-commutative K3 surfaces. On the other hand, an explicit description of a locally complete family of hyperkahler manifolds deformation equivalent to a generalized Kummer variety is not yet available. In this talk we will construct families of non-commutative abelian surfaces as equivariant categories of the derived category of K3 surfaces which specialize to Kummer K3 surfaces. Then we will explain how to induce stability conditions on them and produce examples of locally complete families of hyperkahler manifolds of generalized Kummer deformation type. This is a joint work in progress with Arend Bayer, Alex Perry and Xiaolei Zhao.

Contributed talks

Johannes Horn

Visible Lagrangians of Hitchin systems and square-tiled surfaces

(jt. with Martin Möller and Johannes Schwab) We investigate the question of the existence of visible Lagrangians in Hitchin system. In the $SL(2)$-case we find examples for moduli spaces on certain projective curves, which allow the flat structure of a pillowcase cover. This generalizes to $GL(n)$ for certain projective curves allowing a square-tiled surfaces as spectral cover. These examples are related to subvarieties birational to Hausel's toy model via a Fourier-Mukai transformation.

Andrés Ibáñez Núñez

HKKP theory for algebraic stacks

In work of Haiden-Katzarkov-Konsevich-Pandit (HKKP), a canonical filtration, labeled by sequences of real numbers, of a semistable quiver representation or a vector bundle on a curve is defined. The HKKP filtration is a purely algebraic object, yet it governs the asymptotic behaviour of a natural gradient flow on the space of metrics of the object.

In this talk, we show that the HKKP filtration can be recovered from the stack of semistable objects and a so called norm on graded points. This approach gives a generalisation of the HKKP filtration to other moduli problems of non-linear origin, like GIT quotient stacks and moduli of K-semistable Fano varieties.

Arijit Mukherjee

Symmetric product of curves, tautological algebra and diagonal & weak point property of higher rank divisors

In the first part of this talk, we discuss about the tautological algebra, the algebra generated by the cohomology classes (resp. cycles) of the Brill-Noether subvarieties inside the cohomology ring (resp. Chow ring) of the moduli space of semistable bundles over a curve and also describe its significance. We provide a complete description of the mentioned algebra in genus one case and obtain Poincaré-like relations as in the Jacobian of a curve. In the later part, we talk about the diagonal property and weak point property of higher rank divisors and certain Hilbert schemes associated with those divisors. In the process, we provide an upper bound on the number of such Hilbert schemes (up to isomorphism). Furthermore, we show that the obtained bound is sharp as it is attained by curves of some fixed genus. Throughout the talk, we see thorough and extensive use of the symmetric product of curves and its properties.

Dario Weißmann

A functorial approach to the stability of vector bundles

On a smooth projective curve the locus of stable bundles that remain stable on all etale Galois covers prime to the characteristic defines a big open in the moduli space of stable bundles. In particular, the bundles trivialized on some etale Galois cover of degree prime to the characteristic are not dense - in contrast to a theorem of Ducrohet and Mehta stating that all etale trivializable bundles are dense in positive characteristic. As an application we study the closure of the prime to p etale trivializable vector bundles. This closure is closely related to a stratification of the moduli space of stable vector bundles via their decomposition behaviour on Galois covers of degree prime to the characterstic. We obtain mostly sharp dimension estimates for the closure of the prime to p etale trivializable bundles as well as the decomposition strata.