# Geometria em Lisboa Seminar

## Past sessions

### Global Slodowy slices for moduli spaces of λ-connections

The moduli spaces of Higgs bundles and holomorphic connections both have important affine holomorphic Lagrangian subvarieties, these are the Hitchin section and the space of opers, respectively. Both of these spaces arise from the same Lie theoretic mechanism, namely a regular nilpotent element of a Lie algebra. In this talk we will generalize these parameterizations to other nilpotents. The resulting objects are not related by the nonabelian Hodge correspondence, but by an operation called the conformal limit. Time permitting, we will also discuss their relation to Higher Teichmuller spaces.

### What does a Besse contact sphere look like?

A closed connected contact manifold is called Besse when all of its Reeb orbits are closed (the terminology comes from Arthur Besse's monograph "Manifolds all of whose geodesics are closed", which deals indeed with Besse unit tangent bundles). In recent years, a few intriguing properties of Besse contact manifolds have been established: in particular, their spectral and systolic characterizations. In this talk, I will focus on Besse contact spheres. In dimension 3, it turns out that such spheres are strictly contactomorphic to rational ellipsoids. In higher dimensions, an analogous result is unknown and seems out of reach. Nevertheless, I will show that at least those contact spheres that are convex still "resemble" a contact ellipsoid: any stratum of the stratification defined by their Reeb flow is an integral homology sphere, and the sequence of their Ekeland-Hofer capacities coincides with the full sequence of action values, each one repeated according to a suitable multiplicity. This is joint work with Marco Radeschi.

slides_mazzuchelli.pdf

### Twistor constructions of non-compact hyperkähler manifolds

The talk is based on joint work with Roger Bielawski about twistor constructions of higher dimensional non-compact hyperkähler manifolds with maximal and submaximal volume growth. In the first part of the talk, based on arXiv:2012.14895, I will discuss the case of hyperkähler metrics with maximal volume growth: in the same way that ALE spaces are closely related to the deformation theory of Kleinian singularities, we produce large families of hyperkähler metrics asymptotic to cones exploiting the theory of Poisson deformations of affine symplectic singularities. In the second part of the talk, I will report on work in progress about the construction of hyperkähler metrics generalising to higher dimensions the geometry of ALF spaces of dihedral type. We produce candidate holomorphic symplectic manifolds and twistor spaces from Hilbert schemes of hypertoric manifolds with an action of a Weyl group. The spaces we define are closely related to Coulomb branches of 3-dimensional supersymmetric gauge theories.

### Compact Hyper-Kählers and Fano Manifolds

Projective hyper-Kähler (HK) manifolds are among the building blocks of projective manifolds with trivial first Chern class. Fano manifolds are projective manifolds with positive first Chern class.

Despite the fact that these two classes of algebraic varieties are very different (HK manifolds have a holomorphic symplectic form which governs all of its geometry, Fano manifolds have no holomorphic forms) their geometries have some strong ties. For example, starting from some special Fano manifolds one can sometimes construct HK manifolds as parameter spaces of objects on the Fano. In this talk I will explain this circle of ideas and focus on some recent work exploring the converse: given a projective HK manifold, how to recover a Fano manifold from it?

slides_sacca.pdf

### On the number of fixed points of periodic flows

Finding the minimal number of fixed points of a periodic flow on a compact manifold is, in general, an open problem. We will consider almost complex manifolds and see how one can obtain lower bounds by retrieving information from a special Chern number.

Session slides

### Aspects of abelianization: exact WKB, classical Chern-Simons theory and knot invariants

There is a long history of attacking problems involving nonabelian Lie groups by reducing to a maximal abelian subgroup. It has been understood in the last decade that

1) the exact WKB method for studying linear ODEs,
2) the computation of classical Chern-Simons invariants of flat connections,
3) the study of some link invariants, such as the Jones polynomial,

can all be understood as aspects of this general idea. I will describe this point of view, trying to emphasize the common features of all three problems, and (briefly) their common origin in supersymmetric quantum field theory. Parts of the talk are a report of joint work with Dan Freed, Davide Gaiotto, Greg Moore, Lotte Hollands, and Fei Yan.

### Duality and coproducts in Rabinowitz-Floer homology

The goal of the talk is to explain a duality theorem between Rabinowitz-Floer homology and cohomology. These are Floer homology groups associated to the contact boundary of a Liouville domain, and the duality isomorphism is compatible with canonically defined product structures. Dual to the cohomological product is a homology coproduct which satisfies a remarkable compatibility relation with the product structure. We will also discuss the relationship to loop spaces and Chas-Sullivan/Goresky-Hingston products.

### Towards a polarization-free quantization

We show how a variant of geometric quantization which is free of a choice of polarization may be defined, and also possible problems with this method.

### A Smale-Barden manifold admitting K-contact but not Sasakian structure

Sasakian manifolds are odd-dimensional counterparts of Kahler manifolds in even dimensions, with K-contact manifolds corresponding to symplectic manifolds. In this talk, we give the first example of a simply connected compact 5-manifold (Smale-Barden manifold) which admits a K-contact structure but does not admit any Sasakian structure, settling a long standing question of Boyer and Galicki.

For this, we translate the question about K-contact 5-manifolds to constructing symplectic 4-orbifolds with cyclic singularities containing disjoint symplectic surfaces of positive genus. The question on Sasakian 5-manifolds translates to the existence of algebraic surfaces with cyclic singularities containig disjoint complex curves of positive genus. A key step consists on bounding universally the number of singular points of the algebraic surface.

Muñoz_slides.pdf

### Persistence and Triangulation in Lagrangian Topology

Both triangulated categories as well as persistence homology play an important role in symplectic topology. The goal of this talk is to explain how to put the two structures
together, leading to the notion of a triangulated persistence category. The guiding principle comes from the theory of Lagrangian cobordism.

The talk is based on ongoing joint work with Octav Cornea and Jun Zhang.

### On relations between K-moduli and symplectic geometry

A natural intriguing question is the following: how much the moduli spaces of certain polarized varieties know about the symplectic geometry of the underneath manifold? After giving a general overview, I will discuss work-in-progress with T. Baier, G. Granja and R. Sena-Dias where we investigate some relations between the topology of the moduli spaces of certain varieties, of the symplectomorphism group and of the space of compatible integrable complex structures. In particular, using results of J. Evans, we show that the space of such complex structures for monotone del Pezzo surfaces of degree four and five is weakly homotopically contractible.

Spotti slides.pdf

### Lagrangian fibrations by Prym varieties

Lagrangian fibrations on holomorphic symplectic manifolds and orbifolds are higher-dimensional generalizations of elliptic K3 surfaces. They are fibrations whose general fibres are abelian varieties that are Lagrangian with respect to the symplectic form. Markushevich and Tikhomirov described the first example whose fibres are Prym varieties, and their construction was further developed by Arbarello, Ferretti, and Sacca and by Matteini to yield more examples. In this talk we describe the general framework, and consider a new example. We describe its singularities and show that it is a 'primitive' symplectic variety. We also construct the dual fibration, using ideas of Menet. This is joint work with Chen Shen.

Sawon slides

### Counting curves and stabilized symplectic embedding conjecture

This is a report on joint work with Kyler Siegel that develops new ways to count $J$-holomorphic curves in $4$-dimensions, both in the projective plane with multi-branched tangency constraints, and in noncompact cobordisms between ellipsoids. These curves stabilize, i.e. if they exist in a given four dimensional target manifold $X$ they still exist in the product $X \times {\mathbb R}^{2k}$. This allows us to establish new cases of the stabilized embedding conjecture for symplectic embeddings of an ellipsoid into a ball (or ellipsoid).

McDuff slides.pdf

### On the Yau-Tian-Donaldson conjecture for spherical varieties

I will present how uniform K-stability translates into a convex geometric problem for polarized spherical varieties. From this, we will derive a combinatorial sufficient condition of existence of constant scalar curvature Kahler metrics on smooth spherical varieties, and a complete solution to the Yau-Tian-Donaldson conjecture for cohomogeneity one manifolds.

notes_Thibaut_Delcroix.pdf

### Co-associative fibrations of $G_{2}$-manifolds and deformations of singular sets

The first part of the talk will review background material on the differential geometry of $7$-dimensional manifolds with the exceptional holonomy group $G_{2}$. There are now many thousands of examples of deformation classes of such manifolds and there are good reasons for thinking that many of these have fibrations with general fibre diffeomorphic to a $K3$ surface and some singular fibres: higher dimensional analogues of Lefschetz fibrations in algebraic geometry. In the second part of the talk we will discuss some questions which arise in the analysis of these fibrations and their "adiabatic limits". The key difficulties involve the singular fibres. This brings up a PDE problem, analogous to a free boundary problem, and similar problems have arisen in a number of areas of differential geometry over the past few years, such as in Taubes' work on gauge theory. We will outline some techniques for handling these questions.

Donaldson slides.pdf

Projecto FCT UIDB/04459/2020.

### Dynamical implications of convexity beyond dynamical convexity

We will show sharp dynamical implications of convexity on symmetric spheres that do not follow from dynamical convexity. It allows us to furnish new examples of dynamically convex contact forms that are not equivalent to convex ones via contactomorphisms that preserve the symmetry. Moreover, these examples are $C^1$-stable in the sense that they are actually not equivalent to convex ones via contactomorphisms that are $C^1$-close to those preserving the symmetry. Other applications are the multiplicity of symmetric non-hyperbolic closed Reeb orbits under suitable pinching conditions and the existence of symmetric elliptic periodic Reeb orbits.

This is ongoing joint work with Miguel Abreu.

Macarini slides.pdf

Projecto FCT UIDB/04459/2020.

### Loops in the fundamental group of $\mathrm{Symp}(M,\omega)$ which are not represented by circle actions

It was observed by J. Kędra that there are many symplectic $4$-manifolds $(M, \omega)$, where $M$ is neither rational nor ruled, that admit no circle action and $\pi_1 (\operatorname{Ham}( M))$ is nontrivial. In the case $M={\mathbb C\mathbb P}^2\#\,k\overline{\mathbb C\mathbb P}\,\!^2$, with $k \leq 4$, it follows from the work of several authors that the full rational homotopy of $\operatorname{Symp}(M,\omega)$, and in particular their fundamental group, is generated by circle actions on the manifold. In this talk we study loops in the fundamental group of $\operatorname{Symp}_h({\mathbb C\mathbb P}^2\#\,5\overline{\mathbb C\mathbb P}\,\!^2)$ of symplectomorphisms that act trivially on homology, and show that, for some particular symplectic forms, there are loops which cannot be realized by circle actions. Our work depends on Delzant classification of toric symplectic manifolds and Karshon’s classification of Hamiltonian circle actions.

This talk is based in joint work with Miguel Barata, Martin Pinsonnault and Ana Alexandra Reis.

Silvia slides.pdf

Projecto FCT UIDB/04459/2020.

### Torsion line bundles and branes on the Hitchin system

The locus of the Higgs moduli space fixed under tensorization by a torsion line bundle a key role in the work of Hausel and Thaddeus on topological mirror symmetry. We shall describe the behavior under mirror symmetry of this fixed locus.

Slides of the talk

### The rectangular peg problem

For any smooth Jordan curve and rectangle in the plane, we show that there exist four points on the Jordan curve forming the vertices of a rectangle similar to the given one.

Joint work with Josh Greene.

Projecto FCT UIDB/04459/2020.

### On symplectic inner and outer radii of some convex domains

Symplectic embedding problems are at the heart of the study of symplectic topology. In this talk we discuss how to use integrable systems to compute the symplectic inner and outer radii of certain convex domains.

The talk is based on a joint work with Vinicius Ramos.