Schedule for: 21w5049 - Locally Conformal Symplectic Manifolds: Interactions and Applications (Online)

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

We will begin with the basic definitions of locally conformal symplectic manifolds, justifying the terms "locally" and "conformal" in the name. After understanding the objects of interest (and providing a few examples), we will study isomorphisms and infinitesimal automorphisms, and how basic symplectic tools appear in the locally conformal symplectic theory.

We will review some constructions in symplectic topology, such as the h-principle for overtwisted contact structures, the symplectization of cobordisms and the h-principle for locally constant symplectic forms.

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I shall give an account of the state of the art in LCK geometry, with focus on Vaisman and LCK metrics with potential.

We present several results on the existence of special Hermitian metrics of non-Kähler type on Oeljeklaus-Toma manifolds and relate them to topological and number-theoretical conditions.

We will motivate and introduce one analogue of Weinstein conjecture in lcs geometry, and discuss partial results. This involves some theory of elliptic curves in lcs manifolds, and is related to potential existence of sky catastrophes, which by itself presents a number of interesting open problems. The talk is aimed to be not very technical.

In this talk we describe a number of results on locally conformally symplectic structures of the first kind on compact manifolds. We also study Lie groups endowed with a left-invariant locally conformal symplectic structure of the first kind. Joint work with D. Angella, J.C. Marrero and M. Parton.

We will explain how one can study pseudoholomorphic curves on lcs manifolds in the spirit of the study of contact instantons and describe possible applications thereof. Part of talk is based on a joint work with Yasha Savelyev.

A Hermitian metric on a complex manifold is called strong Kähler with torsion (SKT) or pluriclosed if the torsion of the associated Bismut connection is closed. I will present some general results on SKT metrics in relation to symplectic geometry, the pluriclosed flow and Kähler-like curvature conditions. Moreover, I will discuss a conformal generalization of the SKT condition.

On non-Kähler surfaces it is not always possible to define a locally conformally structure (LCK), however it is always possible to define locally symplectic structures (LCS) which tame the holomorphic structure. From these structures it is possible to derive currents and then PSH functions on the Z-covering. Considering known examples and some Brunella’s theorems we shall see that the GSS conjecture reduces to show the existence of some PSH {\it and PH} functions on the Z-covering.

Floer theory allows to relate the dynamics of Hamiltonian isotopies and the homology of the ambiant symplectic manifold. H.V. Le and K. Ono (as well as J.-C. Sikorav, M. Damian, or A. Gadbled for the Lagrangian version) did generalize Floer ideas to the case of symplectic (non Hamiltonian) isotopies, relating their dynamics to the Novikov homology associated to the flux invariant. I will discuss what remains of this picture regarding the fundamental group: on one hand Floer theory for Hamiltonian isotopies is rich enough to recover generators of the (usual) fundamental group, and on the other hand, each degree 1 cohomolgy class naturally gives rise to a Novikov version of the fundamental group that keeps track of it. The goal of the talk will then be to explain how Floer theory allows to recover genertors of this Novikov fundamental group out of the dynamics of symplectic isotopies when the flux is not too big.

Kato manifolds are compact complex manifolds containing a global spherical shell, i.e. a well embedded sphere which does not disconnect the manifold. They are non-Kähler, but often bear LCK metrics. In this talk, I will first introduce these manifolds and motivate their study by questions arising from LCK geometry. Then I will describe the construction of a particular class of them, using tools of algebraic toric geometry. Finally, I will explain how one can study some of their analytical invariants. This is joint work with A. Otiman, M. Pontecorvo et M. Ruggiero.

(jww M. Bertelson) I shall explain how on every closed real manifold endowed with a nonexact closed 1-form eta, every homotopy class of nondegenerate 2-forms (if any) contains a (locally) conformal symplectic form omega whose Lee class is eta. One can also prescribe the Novikov cohomology class of omega with respect to eta. This "existence h-principle" uses two powerful tools: the Borman-Eliashberg-Murphy h-principle for overtwisted contact structures, and the Eliashberg-Murphy symplectization of cobordisms. Depending on time, I shall give hints of application of this result to the "linear" deformation of foliations into contact structures in large dimensions.

It is well known that on a compact Kähler manifold every conformal vector field is Killing and every Killing vector field is holomorphic. In this talk I will explain how these two results extend to compact locally conformally Kähler manifolds. More precisely, we will see that any conformal vector field on a compact lcK manifold is Killing with respect to a special metric of the conformal class, the so-called Gauduchon metric. Furthermore, any conformal vector field on a compact lcK manifold whose Kähler cover is neither flat, nor hyperkähler, is holomorphic. This talk is based on joint work with Andrei Moroianu.

A nilmanifold is a compact manifold obtained as the quotient of a nilpotent Lie group by a discrete co-compact subgroup. In this talk I will show how the natural filtration of a nilpotent Lie algebra induces a gradation of the de Rham cohomology of a nilmanifold, throught spectral sequences. We will describe necessary conditions for the corresponding nilmanifold to admit symplectic structures in terms of this gradation. Such conditions led us to the classification of symplectic nilmanifolds, when restricted to certain families.