Abstracts

Ugo Bruzzo - Superschemes embeddable into supergrassmannians

In supergeometry  the condition of being projective is more stringent than  in ordinary  algebraic geometry.  For instance,  supergrassmannians often are not projective.  So the techniques,  so often used, of embedding  schemes into projective spaces cannot  always be generalized to supergeometry.   On the  other  hand  an embedding  into  a supergrassmannian is ofter sufficient to obtain the desidered results. In this talk I will discuss this issue and will give an example where the  embedding  into  a  supergrassmannian turns  out  to  be  useful, namely, the representability of the super Hilbert functor.

Fabrizio Catanese -  Spaces of cyclic and Abelian coverings of real Algebraic curves

The Galois coverings of curves, in both the complex and the real case, are related via the Riemann existence theorem to epimorphisms of the fundamental group, respectively of the real fundamental group.

A basic principle is that we obtain connected components of the moduli spaces (called also Hurwitz spaces) once the topological type of the covering is fixed.

The determination of the topological type is difficult for a general group G, but in the complex case one has a nice description in the case where G is Abelian, due to Edmonds, in terms of the local branching invariants, and of a single homological invariant (which in the unramified case lies in the second homology group H_2(G,\ZZ)).

I shall first recall the Comessatti description of the topology of real curves, which divides the real curves into three main types: the  orientable case, the nonorientable case with real points, and the (nonorientable) case without real points.

The first case can be treated in analogy with the complex case, and described again in terms of homological and branching invariants, which are more complicated and relate to  the automorphism M of G with square the identity which corresponds to the real structure of C.

The other two allow a description of the topological types as certain orbits of an infinite group on a set of monodromies.

In particular, I shall illustrate the already wild case without real points where the Klein surface has genus 1, and one can determine such orbits.

[work in progress  with Michael Lönne and Fabio Perroni]

Ben Davison - Nonabelian Hodge theory for stacks

The nonabelian hodge isomorphism provides a homeomorphism of underlying topological (analytic) spaces between the moduli space of semistable Higgs bundles on a smooth projective curve and the moduli space of representations of the fundamental group of the curve.  In particular, whatever your favourite topological invariant is, it is the same for these two moduli spaces.

Passing to the stacks of semistable Higgs bundles and representations of the fundamental group, we no longer have such a homeomorphism.  I will explain how it is possible nonetheless to show that these stacks have isomorphic Borel-Moore homology, by showing that they have isomorphic BPS cohomology (which I'll define!).  This, in turn, is achieved via a general theorem relating BPS cohomology of stacks in "totally negative" 2-Calabi-Yau categories to intersection cohomology of coarse moduli spaces.  This is joint work with Lucien Hennecart and Sebastian Schlegel Mejia

Barbara Fantechi - Moduli spaces of generalized syzygy bundles

This is a report on recent work in collaboration with RM Miró Roig.

Let F be a vector bundle on a smooth projective variety X of dim n>1; if F is generated by global sections, its syzygy bundle is the kernel of the map from the trivial bundle with fiber H^0(F) to F. A generalized syzygy bundle S=S_{F,W} is the same when W in H^0(F) is a subspace generating F at every point.

We show that under mild assumptions, the natural map from the moduli space of gen syzygy bundles to the moduli of simple bundles on X is a locally closed embedding. If moreover q(X)=0 we show that under one extra assumption (also mild if n>2) it is in fact an open embedding.

Camilla Felisetti - On the intersection cohomology of vector bundles

Intersection cohomology is a topological notion adapted to the description of singular topological spaces, and the Decomposition Theorem for algebraic maps is a key tool in the subject. The study of the intersection cohomology of the moduli spaces of semistable bundles on Riemann surfaces began in the 80’s with the works of Frances Kirwan. Motivated by the work of Mozgovoy and Reineke, in joint work with Andras Szenes and Olga Trapeznikova, we give a complete description of these structures via a detailed analysis of the Decomposition Theorem applied to a certain map from parabolic bundles. We also give a new formula for the intersection Betti numbers of these moduli spaces, which has a clear geometric meaning. In the talk, I will give an introduction to the subject, and describe our results.

Soheyla Feyzbakhsh - Explicit formulae for rank zero DT invariants

Fix a Calabi-Yau 3-fold X of Picard rank one satisfying the Bogomolov-Gieseker conjecture of Bayer-Macr\`i-Toda, such as the quintic 3-fold. I will first describe explicit formulae relating rank zero Donaldson-Thomas (DT) invariants to Pandharipande-Thomas (PT) invariants using wall-crossing with respect to weak Bridgeland stability conditions on X. As applications, I will find sharp Castelnuovo-type bounds for PT invariants, and explain how combining these explicit formulae with S-duality in physics enlarges the known table of Gopakumar-Vafa (GV) invariants. The second part is joint work with string theorists Sergei Alexandrov, Albrecht Klemm, Boris Pioline and Thorsten Schimannek. 

Alessandro Giacchetto - On the spin GW/Hurwitz correspondence

Spin Gromov–Witten invariants were introduced by Kiem and Li as a tool to determine the ordinary Gromov–Witten invariants of surfaces with smooth canonical divisors. Conjecturally, spin Gromov–Witten invariants are given by linear combinations of spin Hurwitz numbers, which in turn can be computed via representation theory. This is the so-called spin GW/Hurwitz correspondence. In this talk, I will present a proof of this conjecture for target P^1, and explain how the general case follows from a conjectural degeneration formula for spin GW invariants. Joint work with R. Kramer, D. Lewański, A. Sauvaget.

Lothar Göttsche - (Refined) Verlinde and Segre formulas for Hilbert schemes of points 

This is joint work with Anton Mellit. Segre and Verlinde numbers of Hilbert schemes of points have been studied for a long time. The Segre numbers are evaluations of top Chern and Segre classes of so-called tautological bundles on Hilbert schemes of points. The Verlinde numbers are the holomorphic Euler characteristics of line bundles on these Hilbert schemes. We give the generating functions for the Segre and Verlinde numbers of Hilbert schemes of points. The formula is proven for surfaces with K_S^2=0, and conjectured in general. Without restriction on K_S^2 we prove the conjectured Verlinde-Segre correspondence relating Segre and Verlinde numbers of Hilbert schemes. Finally we find a generating function for finer invariants, which specialize to both the Segre and Verlinde numbers, giving some kind of explanation of the Verlinde-Segre correspondence. 

Victoria Hoskins - Motivic mirror symmetry for Higgs bundles

Moduli spaces of Higgs bundles for Langlands dual groups are conjecturally related by a form of mirror symmetry. For SL_n and PGL_n, Hausel and Thaddeus conjectured a topological mirror symmetry given by an equality of (twisted orbifold) Hodge numbers, which was proven by Groechenig--Wyss--Ziegler and later by Maulik--Shen. We lift this to an isomorphism of Voevodsky motives, and thus in particular an equality of (twisted orbifold) rational Chow groups. Our method is based on Maulik and Shen's approach to the Hausel--Thaddeus conjecture, as well as showing certain motives are abelian, in order to use conservativity of the Betti realisation on abelian motives. This is joint work with Simon Pepin Lehalleur.

Martijn Kool - Surfaces on Calabi-Yau fourfolds

Building on work of Oh-Thomas, I will introduce invariants for counting surfaces on Calabi-Yau fourfolds. In a family, they are deformation invariant along Hodge loci. If non-zero, the variational Hodge conjecture for the family under consideration holds. Joint work with Y. Bae and H. Park.

Michael Lönne - Fundamental groups of trigonal strata of abelian differentials
We consider the projectivized loci of abelian differentials where the curve is trigonal and the canonical divisor is multiple of a fibre of a trigonal map. The fundamental group of such loci is shown to have a particularly nice presentation. It gives some new insight into the largely unsolved problem to understand the fundamental groups of moduli spaces of abelian differentials, which have been conjectured by Kontsevich to be commensurate to mapping class groups.

Margarida Melo - Tropicalization of the universal Jacobian: logarithmic and non-archimedian view points

Moduli spaces of tropical objects can often be obtained as tropicalization of suitable compactifications of algebro-geometric objects.

In the talk we will start by giving an overview of the construction of universal compactified Jacobians along with their tropical counterparts, in the category of cone stacks.

We then show that this cone stack can be obtained  by tropicalizing two versions of the algebraic universal Jacobian: the logarithmic universal Jacobian and the non archimedian universal Jacobian. We then discuss, within this picture, new stability conditions for (universal) compactified Jacobians.

The talk will be based on joint work of the speaker with S. Molcho, M. Ulirsch, F. Viviani and J. Wise.

Riccardo Moschetti - The non-degeneracy invariant of Enriques surfaces

The geometry and combinatorics of elliptic fibrations on Enriques surfaces is fairly well understood for general Enriques surfaces (i.e., Enriques surfaces without smooth rational curves, also called unnodal). However, the behavior of elliptic fibrations changes radically when we specialize our Enriques surface. The non-degeneracy invariant is the maximum number of half-fibers pairwise intersecting giving 1, where a half-fiber is a curve F on the surface such that |2F| is an elliptic pencil. This invariant encodes many relations about the geometry of an Enriques surfaces, and of its derived category. I will talk about a series of ongoing works joint with Franco Rota and Luca Schaffler, concerning the computation of this invariant in some non-general cases. 

Paolo Rossi - Counting meromorphic differentials on the Riemann sphere

How many meromorphic differentials are there, on the Riemann sphere, up to Mobius transformations? This problem has a finite answer in three cases only: when there are two poles and no zeros, one zero and two poles, or one pole and two zeros. Each of these cases can be enriched to an infinite sequence by allowing any number of extra residueless poles. I will present the explicit solution to these counting problems we computed with A. Buryak. It uses surprisingly non-elementary techniques of intersection theory in the moduli spaces of differentials and, in particular, a suitable generalization of the WDVV relations in the cohomology of the moduli space of stable curves. It also relates this problem with the KP hierarchy.

Orsola Tommasi - Geometry of fine compactified Jacobians
The degree d universal Jacobian parametrizes degree d line bundles on smooth curves. There are several approaches on how to extend it to a proper family over the moduli space of stable curves. In this talk, we introduce a simple definition of a fine compactified universal Jacobian, both for a single nodal curve and for families. We discuss their combinatorial characterization and explain how this leads to new examples already in the case of curves of genus 1. This is joint work with Nicola Pagani and Jesse Kass.