# Working Seminar on Symplectic/Contact Geometry/Topology

## Past sessions

### From semi-toric systems to Hamiltonian ${S}^{1}$-spaces

Semi-toric integrable systems on closed four dimensional manifolds, introduced by Vu Ngoc, lie at the intersection of integrable Hamiltonian systems and Hamiltonian torus actions. In particular, they are integrable Hamiltonian systems which have one integral that generates an effective Hamiltonian ${S}^{1}$-action. Vu Ngoc showed that these systems share an important property with symplectic toric manifolds, i.e. it is possible to associate a family of convex polygons to each of them. On the other hand, considering the manifold only with the Hamiltonian ${S}^{1}$-action, they give rise to examples of Hamiltonian ${S}^{1}$-spaces, which have been classified by Karshon. The aim of this talk is to illustrate how, given a semi-toric system, Karshon's invariants of the underlying Hamiltonian ${S}^{1}$-space can be reconstructed from one (and hence any) convex polygon associated to the system. Time permitting, we will consider how to construct examples of these systems starting from symplectic toric manifolds.

This is joint work with Sonja Hohloch and Silvia Sabatini.

### Singular integral affine structures and integrable Hamiltonian systems

An important outstanding question in the theory of integrable Hamiltonian systems is their classification, i.e. the construction of some (hopefully computable!) invariants which completely determine these systems up to a suitable notion of equivalence. In this talk, a weaker (but still considerably hard) problem is going to be introduced, namely that of determining when two integrable Hamiltonian systems are equivalent. In the absence of singularities, i.e. equilibria, this problem can be solved using integral affine geometry, which studies symmetries of Euclidean space fixing the standard integral lattice therein; this was first observed by Duistermaat and Dazord and Delzant. One way to extend this result to include singularities is to define an appropriate notion of singular integral affine geometry, which is the aim of this talk. Time permitting, some low dimensional explicit examples will be discussed. This is joint ongoing work with Rui Loja Fernandes.

### On complete isotropic realisations of Poisson manifolds

Poisson geometry can be viewed as the study of those manifolds which admit a (possibly singular) foliation by symplectic manifolds (e.g. $$\mathbb{R}^3$$ foliated by spheres centred at the origin of increasing radius and the radius). A complete symplectic realisation of a Poisson manifold is a desingularisation, in the sense that it consists of a symplectic manifold together with a submersion onto the original Poisson manifold which reflects its Poisson structure. A beautiful theorem of Crainic and Fernandes proves that existence of such a desingularisation is equivalent to the Poisson manifold (viewed as an infinitesimal object, the analogue of a Lie algebra) admitting a global integration (the analogue of a Lie group). In this talk, we consider complete isotropic realisations of Poisson manifolds, which are desingularisations as above of minimal dimension; these are related to non-commutative integrable Hamiltonian systems. The fundamental driving question behind this talk is the following: what can be said about Poisson manifolds that admit such desingularisations? Some results in this direction, both old and new, will be presented. This is ongoing joint work with Ioan Marcut.

### Lecture VII - Lagrangian fibrations with elliptic singularities

The Eliasson-Miranda-Zung linearisation theorem provides a symplectic model in a neighbourhood of a compact non-degenerate orbit of a completely integrable Hamiltonian system. A natural question to ask is whether such a model exists for singular fibres of Lagrangian fibrations (which, generally, consist of several disjoint orbits), the aim being a symplectic classification of Lagrangian fibrations with singularities. In this lecture the simplest case of purely elliptic singularities is studied in order to illustrate some of the difficulties that arise in developing such a classification theory.

### Lecture VI - The Eliasson-Miranda-Zung linearisation theorem

The definition of non-degenerate singular orbits of completely integrable Hamiltonian systems gives local linear models for the underlying Hamiltonian ${ℝ}^{n}$-action. A natural question is whether there exists a symplectomorphism from an open neighbourhood of a fixed singular orbit to the local linear model which preserves the respective Lagrangian fibrations. The Eliasson-Miranda-Zung theorem answers the above questions affirmatively when the orbit is compact. In this lecture this theorem is stated and a strategy for its proof is outlined, so as to explain the main underlying geometric ideas.

### Lecture V - Non-degenerate singularities

Having established several structural results for Lagrangian fibrations which are submersions in the first four sessions, this lecture aims to study a more general class of fibrations which allows for topologically well-behaved (i.e. Morse-Bott in some sense) singularities. These naturally arise in the theory of completely integrable Hamiltonian systems (e.g. if the phase space is compact, they must exist) and in mirror symmetry. The notion of non-degenerate singularities is introduced and illustrated with several examples. Time permitting, the Eliasson-Miranda linearization theorem for non-degenerate singularities is going to be studied in some detail (but probably without proof).

http://www.math.ist.utl.pt/~jmourao/inves/D_Sepe_Lagrangian_Fibrations.pdf

### Lecture IV - (Integral) affine manifolds and Lagrangian fibrations II

This lecture continues with the study of (integral) affine manifolds. First, some important invariants associated to these manifolds (the affine and linear holonomies and the radiance obstruction) are constructed. Particular attention is devoted to the radiance obstruction, a cohomology class constructed by Goldman and Hirsch which contains important information about the given (integral) affine structure. Secondly, it is proved that a manifold is the base of any Lagrangian fibre bundle with compact and connected fibres if and only if it is an integral affine manifold, which allows to study the problem of constructing all Lagrangian fibre bundles with compact and connected fibres over a fixed integral affine manifold. As usual, the theory developed is illustrated by means of examples.

http://www.math.ist.utl.pt/~jmourao/inves/D_Sepe_Lagrangian_Fibrations.pdf

### Lecture III - (Integral) affine manifolds and Lagrangian fibrations I

A theorem due to Weinstein states that the leaves of any Lagrangian foliation admit a flat, torsion-free connection. A manifold admitting such a connection is called affine; these have been extensively studied since the '50s as a generalisation of flat Riemannian manifolds. This lecture first proves the above result directly for Lagrangian fibrations and then introduces more formally (integral) affine manifolds, illustrating the theory with examples.

http://www.math.ist.utl.pt/~jmourao/inves/D_Sepe_Lagrangian_Fibrations.pdf

### Lecture II - Topological and symplectic classification

A theorem due to Liouville, Mineur and Arnol'd states that if a Lagrangian fibration admits a compact and connected fibre $F$, then the fibre is diffeomorphic to a torus, nearby fibres are also tori and there exists a symplectomorphism between a neighbourhood of $F$ and the zero section of the cotangent bundle to the torus which preserves the fibrations. In this lecture, a generalisation of this theorem for complete Lagrangian fibrations is proved, using the natural fibrewise action of the cotangent bundle to the base on the total space of the fibration. This construction allows to develop a topological (in fact, smooth) and symplectic classification theory for such fibrations, which yields two topological invariants, the period net and Chern class, and one symplectic characteristic class, the Lagrangian Chern class.

http://www.math.ist.utl.pt/~jmourao/inves/D_Sepe_Lagrangian_Fibrations.pdf

### Lecture I - An introduction to Lagrangian fibrations

This lecture presents an overview on the whole series, introducing some fundamental concepts in the study of Lagrangian fibrations by means of some examples. Special attention is dedicated to completely integrable Hamiltonian systems, which are a type of mechanical systems that are intimately connected to Lagrangian fibrations. The structure of the cotangent bundle as a Lagrangian fibration (in fact a fibre bundle) is studied, thus illustrating the ideas that lead to the topological and symplectic classification of Lagrangian fibrations.

http://www.math.ist.utl.pt/~jmourao/inves/D_Sepe_Lagrangian_Fibrations.pdf

### Hyperkähler manifolds - II

Hypertoric manifolds are $4n$ dimensional hyperkähler manifolds admiting a tri-hamiltonian ${𝕋}^{n}$ action. We describe a construction of such manifolds as quotients of ${ℂ}^{2d}$ by subtori of ${𝕋}^{d}$. This construction mimics Delzant's construction of toric Kahler manifolds as quotients of ${ℂ}^{d}$ by subtori of ${𝕋}^{d}$. We will discuss relations between the two constructions and explain how to use the hyperkahler version to give a description of the hyperkähler metric in hyper action-angle coordinates.

http://www.math.ist.utl.pt/~jmourao/ws/HK.pdf

### Hyperkahler manifolds - I

##### Motivational introduction
Q1
Why would a differential/symplectic/algebraic geometer care about hyperkähler manifolds?
Q2
Why would a quantum geometer care?
##### Technical introduction

Definition and basic properties of hyperkähler manifolds. Relation with the holomorphic symplectic point of view. Hyperkähler quotient/holomorphic symplectic quotient as a tool to construct interesting examples.

http://www.math.ist.utl.pt/~jmourao/ws/HK.pdf

### Symplectic quasi-states and intersections - I

Current organizer: Miguel Abreu

a unit of the Associate Laboratory LARSyS