1999 seminars

Holger Spielberg, CAMGSD-IST
Quantum cohomology and symplectic cuts

The quantum cohomology ring of a symplectic manifold is a deformation of the usual cohomology ring. Since its structure constants are given by Gromov-Witten invariants, the quantum ring, too, is an invariant of the symplectic topology of the manifold. In this talk I will report on recent progress (by Li/Ruan, Ionel/Parker, and others) on the question how this quantum ring changes under symplectic cuts (or gluing, which is the inverse operation). In the case of Kähler manifolds, these operations include for example the blow-up along complex submanifolds.

Yvette Kosmann-Schwarzbach, École Polytechnique
Quasi-Poisson actions

We first introduce quasi-Poisson actions and their momentum maps, which take values in homogeneous spaces. We then show that Poisson actions of Poisson Lie groups (when the Manin pair under consideration is in fact a Manin triple) and the Hamiltonian actions with group-valued momentum maps (when the Manin pair has vanishing cobracket) are special cases of this construction.

António Araújo, CMAF, Universidade de Lisboa
Moduli Spaces of germs of Legendrian Curves

We construct the moduli spaces of germs of conic singular Lagrangean subvarieties of a symplectic manifold of dimension four. The projectivisation of a Lagrangean variety is a Legendrian subvariety of a contact manifold.

Sílvia Anjos, Instituto Superior Técnico
Topology of the symplectomorphism groups of $S^2\times S^2$

In this talk we discuss the topology of the symplectomorphism group of a product of two $2$-dimensional spheres where the ratio of the areas of the spheres is bigger than $1$. We compute the homotopy type of this symplectic manifold and we how that the group of symplectomorphisms contains two finite dimensional Lie groups generating the homotopy.

Rita Gaio, Universidade do Porto
J-holomorphic curves, moment maps and adiabatic limits

We study pseudoholomorphic curves in symplectic quotients as adiabatic limits of solutions of a system of nonlinear first order elliptic partial differential equations in the ambient symplectic manifold. The symplectic manifold carries a Hamiltonian group action. The equations involve the Cauchy-Riemann operator over a Riemann surface, twisted by a connection, and couple the curvature of the connection with the moment map. Our work should prove a conjecture that the genus zero invariants of Hamiltonian group actions defined by these equations agree with the genus zero Gromov-Witten invariants of the symplectic quotient in the monotone case.

Le Dung-Trang, Université de Provence
Resolution of surfaces by normalized blow-ups

A celebrated theorem of O. Zariski asserts that if $S$ is an algebraic surface over an algebraically closed field of characteristic zero one may resolve the singularities of $S$ by a finite sequence of point blow-ups followed by normalizations. We present a new approach to the proof of this theorem.

Pedro Cristiano, CMAF, Universidade de Lisboa
On a Class of Local Systems Associated to Irreducible Plane Curves

We study a class of local systems on the complementary of an irreducible plane curve. We exhibit families of such local systems which by microlocal Riemann-Hilbert correspondence,give rise to regular holonomic $D$-modules with characteristic variety equal to the union of the zero section with the conormal of the curve.

Miguel Abreu, Instituto Superior Técnico
Kahler metrics on toric orbifolds

In this talk I will describe how a symplectic approach to Kahler geometry on toric manifolds, presented in this seminar on July 2000, can be extended to the more general context of toric orbifolds. The main result is again an effective parametrization of all invariant Kahler metrics on a toric orbifold, via smooth functions on the corresponding moment polytope.

As an application I will give a simple explicit description of recent work of R. Bryant, producing interesting new families of extremal Kahler metrics. In particular, one obtains an extremal Kahler metric on any weighted projective space.

José Basto Gonçalves, Universidade do Porto
Linearization of resonant vector fields

Classical linearization results hold in the non-resonant case. For resonant vector fields, we show how the non-linear part either gives simple, geometric, conditions for linearization, or else provides information on which resonant terms are relevant for its normal form.

Iakovos Androulidakis, Instituto Superior Técnico
Geometric quantization and integrability of Lie algebroids

A classical theorem of Kostant gives the conditions for a symplectic form to be the curvature form of a Hermitian connection on a Hermitian line bundle.

We consider a generalization of this result and explain how it relates to the integrability of Lie algebroids and the classification of extensions of Lie groupoids. Also, we clarify the nature of the obstructions using the notion of a crossed module.

Orlando Neto, CMAF, Universidade de Lisboa
On the de Rham complex of an holonomic $\cal D$-module

An holonomic $\mathcal{D}$-module $\mathcal{M}$ is a system of differential equations on a complex manifold $X$ (module over the ring of differential equations on $X$) that equals a flat connection $(E,\nabla )$ on the complementary of an hypersurface $Y$ of $X$. If $\mathcal{M}$ verifies a reasonable condition the de Rham complex of $\mathcal{M}$ only depends on the the Rham complex of $(E,\nabla )$. We obtain in this way a dictionary between a certain class of linear representations of the fundamental group of $X\setminus Y$ and a certain class of holonomic $\mathcal{D}$-modules.

Paul Norbury, Adelaide University
Boundary algebras of hyperbolic monopoles

We prove the conjecture that a monopole in hyperbolic space can be completely determined by its “holographic image” on the conformal boundary two-sphere. The main tool used is an $n$-point function defined for a given monopole and any ordered collection of points on the conformal boundary two-sphere. The n-point function can be used to define structure coefficients of an interesting associative algebra abstractly generated by points of the conformal boundary two-sphere.