# Planned seminars

## 25/01/2022, Tuesday, 16:30–17:30 Europe/Lisbon — Online

Giancarlo Urzua, Pontificia Universidad Católica de Chile

Together with Javier Reyes, in https://arxiv.org/abs/2110.10629 we have been able to construct compact 4-manifolds $3\mathbb{CP}^2\#(19-K^2)\overline{\mathbb{CP}}^2$ with complex structures for $K^2=1,2,3,4,5,6,7,8,9$. The cases $K^2=7,9$ are completely new in the literature, and this finishes with the whole range allowed by the technique of Q-Gorenstein smoothing (rational blow-down). But one can go further: Is it possible to find minimal exotic $3\mathbb{CP}^2\#(19-K^2)\overline{\mathbb{CP}}^2$ for $K^2\geq10$? Here it would be much harder to prove the existence of complex structures, but, as a motivation, there is not even one example for $K^2 > 15$, and very few for $10 \leq K^2 \leq 15$ (see e.g. works by Akhmedov, Park, Baykur). In this talk I will explain the constructions in connection with the geography of spheres arrangements in $K3$ surfaces, where the question of the title arises. We do not have an answer. So far we have been implementing what we know in computer searches, finding these very rare exotic surfaces for $K^2=10,11,12$. This is a new and huge world which promises more findings, we have explored very little.

## 01/02/2022, Tuesday, 16:30–17:30 Europe/Lisbon — Online

Johannes Horn, U. Frankfurt
To be announced

## 08/02/2022, Tuesday, 16:30–17:30 Europe/Lisbon — Online

Jonny Evans, University of Lancaster

(Joint with Y. Lekili) If someone gives you a variety with a singular point, you can try and get some understanding of what the singularity looks like by taking its “link”, that is you take the boundary of a neighbourhood of the singular point. For example, the link of the complex plane curve with a cusp $y^2 = x^3$ is a trefoil knot in the 3-sphere. I want to talk about the links of a class of 3-fold singularities which come up in Mori theory: the compound Du Val (cDV) singularities. These links are 5-dimensional manifolds. It turns out that many cDV singularities have the same 5-manifold as their link, and to tell them apart you need to keep track of some extra structure (a contact structure). We use symplectic cohomology to distinguish the contact structures on many of these links.

## 15/02/2022, Tuesday, 16:30–17:30 Europe/Lisbon — Online

Richard Hind, University of Notre Dame
To be announced

## 22/02/2022, Tuesday, 16:30–17:30 Europe/Lisbon — Online

Louis Ioos, Max Planck Institute for Mathematics (Bonn)
To be announced

## 08/03/2022, Tuesday, 16:30–17:30 Europe/Lisbon — Online

Ana Peón-Nieto, U. Birmingham
To be announced

## 15/03/2022, Tuesday, 16:30–17:30 Europe/Lisbon — Online

Joel Fine, Université Libre de Bruxelles
To be announced

## 05/04/2022, Tuesday, 16:30–17:30 Europe/Lisbon — Online

Christian Pauly, Université de Nice Sophia-Antipolis

A very stable vector bundle over a curve is a vector bundle having no non-zero nilpotent Higgs fields. They were introduced by Drinfeld and studied by Laumon in connection with the nilpotent cone of the Hitchin system. According to Drinfeld non-very stable bundles, also called wobbly bundles, form a divisor in the moduli space of vector bundles. In this talk I will try to explain the motivations for studying the properties of wobbly divisors, with a special focus on the rank-2 (joint work with S. Pal) and rank-3 case (joint work with A. Peon-Nieto).