1999 seminars

Brian Hall, Notre Dame
Complex structures and magnetic fields

If $M$ is a real-analytic Riemannian manifold, then there exists a canonical "adapted complex structure" on a neighborhood of the zero-section in the tangent bundle $T M$. These structures were introduced independently by Guillemin and Stenzel and by Lempert and Szoke. It is possible to understand the adapted complex structure in terms of the "imaginary time geodesic flow." I will describe a new family of complex structures in which one modifies the problem by adding a "magnetic field" on $M$, described by a closed 2-form. These new complex structures are described in terms of the Hamiltonian flow for a particle in a magnetic field, evaluated (in a suitably interpreted way) at an imaginary time. For a constant magnetic field on $\mathbb{R}^2$ or $S^2$, the magnetic complex structure can be computed explicitly and is defined on the entire tangent bundle.

Xavier Roulleau, CAMGSD/IST
Quotients of Fano surfaces

Fano surfaces parametrize the lines on smooth cubic threefolds. In this talk, we will explain the remarkable geometric properties of these surfaces and we will explain how to use these properties in order to compute the invariants of the surfaces obtained as quotients of Fano surfaces by an automorphism group.

Matias del Hoyo, CAMGSD/IST
Vector bundles over Lie groupoids and algebroids

In a real vector space addition is settled by scalar multiplication. This lead us to the following description of smooth vector bundles: they are manifolds endowed with a nice action of the multiplicative monoid of the real numbers. In this talk I will deal with vector bundles over Lie groupoids and algebroids. Lie groupoids are presentations for singular smooth spaces, constitute a concrete unified framework to deal with manifolds, Lie groups, actions, foliations and others. Lie algebroids are their infinitesimal counterpart, together they play a rich theory in actual development. Vector bundles over groupoids and algebroids are generalized representations and help us to understand the geometry of our singular spaces. I will present definitions and examples, provide a characterization by using the multiplicative monoid, and discuss the integration of these structures. This is part of a joint work with H. Bursztyn and A. Cabrera.

Sheila Sandon, CNRS and Institute for Advanced Study
On translated points of contactomorphisms

The Arnold conjecture in Symplectic Topology states existence of many fixed points for Hamiltonian symplectomorphisms of a compact symplectic manifold. In my talk I will discuss an analogue of this conjecture in Contact Topology, based on the notion of translated points.

Éveline Legendre, Université Paul Sabatier, Toulouse
Commonness of Kähler-Einstein metrics in toric geometry

I will explain why each integral convex compact polytope is the moment polytope of a Kähler-Einstein orbifold (unique up to a dilatation). More generally, I will explain how to prove that each convex compact polytope admits a unique ray of symplectic potentials solving the toric Kähler-Einstein equation. Then, I'll bring out some geometric applications.

Ciro Ciliberto, Università di Roma II
Gonality of nodal curves on general K3 surfaces

This is a report on joint work in collaboration with A. Knutsen (University of Bergen, Norway).

The pairs (S,H) with S a complex K3 surface (i.e. S is simply connected and Ω S 2 is trivial), and H is an ample line bundle of genus p (i.e. H 2=2p2), have an irreducible moduli space 𝒦 p of dimension 19. Let (S,H)𝒦 p be general. The linear system H has dimension p and its general element is a smooth curve of genus p. I consider V H,δ, the Severi variety of δ-nodal curves in H, which is locally closed of codimension δ in H and its general element is an irreducible curve with exactly δ nodes and no other singularities, so that its geometric genus (i.e. the genus of its desingularization) is g=pδ. The image of V H,δ in g, the moduli space of curves of genus g, has dimension g and its importance in moduli problems is well known. For any integer k2, consider the subscheme V H,δ kV H,δ given by {CV H,δ: the normalization of C is a k-tuple cover of 1}.

The first question is: when is V kV H,δ not empty? Using a (by now classical) vector bundle technique due to Lazarsfeld, we show that a necessary condition is δα(g(k1)(α+1)) where α:=g2(k1). A dimension count shows that, if V H,δ k is not empty, its expected dimension should be 2k2. As a matter of fact, we prove that for all even p and all δ,k verifying the above relation, V H,δ k is actually not empty, of dimension 2k2. For p odd we have a slightly weaker result. Finally, I will talk about some interesting relations of these results with a conjecture by Hassett and Tschinkel on the Mori cone of the hyperkhähler manifold Hilb k(S) (which parametrizes subschemes of S of finite lenght k).

André Neves
Willmore conjecture and min-max methods

In 1965, Willmore conjectured that for every torus the integral of the square of the mean curvature is bigger or equal to 2π 2. I will explain how to prove the conjecture using min-max methods. This is joint work with Fernando Marques (IMPA).

Simone Marchesi, Univ. Complutense de Madrid
Steiner and Schwarzenberger bundles on Grassmannians

In this talk we will present Steiner and Schwarzenberger bundles on the Grassmannians and show the theory that relates them. In the first part we will define the two families of bundles mentioned before and study their properties. We will introduce then the concept of jumping pair for a Steiner bundle and we will study the dimension of the jumping locus of the bundle. Finally, we will give a complete classification of Steiner bundles on the Grassmannian whose jumping locus has maximal dimension and will describe them as Schwarzenberger bundles.

Enrique Arrondo, Univ. Complutense de Madrid
Subvarieties of small codimension

We intend to give an elementary talk to explain why (smooth) subvarieties of small codimension are expected to be quite special. We will concentrate on a theorem by Barth stating that a subvariety of dimension $n$ in a projective space of dimension $N$ inherits much of the topology of the projective space, namely the integral cohomology up to order $2n-N$ must be the same. We will give a new geometrical approach to this theorem, which will allow us to extend Barth's theorem to other ambient spaces different from the projective space. We will put all this in relation with the famous Hartshorne's conjecture about subvarieties of small codimension in the projective space

Nuno M Romão, Max-Planck-Institut für Mathematik, Bonn
Vortices and Jacobian varieties

Gauged vortices are configurations of fields for certain gauge theories on fibre bundles over a surface $S$. Their moduli spaces support natural $L^2$-metrics, which are Kaehler, and whose geodesic flow approximates vortex dynamics at low speed. My talk will focus on vortices in line bundles, for which the moduli spaces are modelled on the spaces of effective divisors on $S$ with a fixed degree $k$; I shall describe the behaviour of the underlying $L^2$-metrics in a "dissolving limit" where the $L^2$-geometry simplifies and can be related to the geometry of the Jacobian variety of the surface. Some intuition about multivortex dynamics in this limit will be provided by analysing the simplest nontrivial example (two dissolving vortices moving on a hyperelliptic curve of genus three). This is joint work with N. Manton.

Leonor Godinho, Instituto Superior Técnico
Polygons in Minkowski 3-space: A quest for a long-lost family

We consider moduli spaces of polygons in Minkowski 3-space with a fixed number of sides lying in the future time-like cone and the others in the past (where each side has a fixed Minkowski length) and study their relation with the fixed-point set of a natural involution on the space of rank-2 parabolic Higgs bundles over CP 1.

This is joint work with I. Biswas, C. Florentino and A. Mandini.

Ethan Cotterill, Universidade de Coimbra
Some Brill-Noether theory on curves and metric graphs

Brill-Noether theory seeks to answer the basic question of when an abstract complex curve C of genus g comes equipped with a nondegenerate degree-d morphism to P r.

When C is general in the moduli space M g, an answer is provided by a celebrated theorem of Griffiths and Harris, which establishes that C's behavior is as "expected" whenever ρ=g(r+1)(gd+r) is positive, and that there are no morphisms when ρ is negative.

Payne, et al. recently gave a proof of the nonexistence statement using a combinatorial analysis of metric graphs of a particular combinatorial type. These belong to a moduli space M g trop for stable metric graphs, which admits a stratification that is dual to that of the Deligne-Mumford space of stable curves.

We will discuss work in progress with Melo, Neves, and Viviani aimed at understanding the Brill-Noether-type behavior of metric graphs of other combinatorial types, focusing on a remarkable infinite family of these.

Richard Wentworth, University of Maryland
Gluing formulas for determinants of Dolbeault laplacians on Riemann surfaces

We present gluing formulas for zeta regularized determinants of Dolbeault laplacians on Riemann surfaces. These are expressed in terms of determinants of associated operators on surfaces with boundary satisfying local elliptic boundary conditions. The conditions are defined using the additional structure of a framing, or trivialization of the bundle near the boundary. An application to the computation of bosonization constants follows directly from these formulas.

Sean Lawton, University of Texas-Pan American
Topology of Character Varieties of Abelian Groups

Let $K$ be a compact Lie group, $G$ be its complexication, and $F$ be any finitely generated Abelian group. We prove that the conjugation orbit space $\operatorname{Hom}(F,K)/K$ is a strong deformation retract of the GIT conjugation orbit space $\operatorname{Hom}(F,G)/G$. As a corollary, we determine necessary and sufficient conditions for $\operatorname{Hom}(F,G)/G$ to be irreducible when $G$ is connected and semisimple, and $F$ is free Abelian. This is joint work with C. Florentino.

Marta Batoreo, UC Santa Cruz
On the Rigidity of the Maslov Index for Coisotropic Submanifolds

A rigidity result on the coisotropic Maslov index states that there exists a non-trivial loop (tangent to the characteristic foliation of a stable coisotropic submanifold) with certain bounds on its symplectic area and its Maslov index. This was proved by Ginzburg for symplectically aspherical ambient manifolds. The result also holds for some symplectic manifolds not necessarily aspherical. We shall state the theorem for the “rational case” and sketch its proof.

Stavros Papadakis
Equivariant degenerations of spherical modules

Given a reductive algebraic group G and an invariant Hilbert function h, Alexeev and Brion have defined a moduli scheme M which parametrizes affine G-schemes X with the property that the coordinate ring of X decomposes, as G-module, according to the function h. The talk will be about joint work with Bart Van Steirteghem (New York) which studies the moduli scheme M under some additional assumptions.

Allen Knutson, Cornell
Poisson $T$-manifolds with finitely many $T$-leaves

There is a natural holomorphic Poisson structure on any flag manifold $G/P$ with infinitely many symplectic leaves, but only finitely many $T$-orbits of symplectic leaves. I'll describe these in detail for the case of the Grassmannian, where the leaves are naturally indexed by juggling patterns (this work is joint with Thomas Lam and David Speyer). Then I'll talk about manifolds with atlases consisting of Kac-Moody Bruhat cells, which conjecturally includes wonderful compactifications of groups.

Sumio Yamada, Tohoku University, Japan
Teichmuller space and convex geometry

In this talk, I will give a survey in convex geometries associated with Teichmuller space. In particular, an emphasis is put on the Weil-Petersson geometry, where deformations of hyperbolic metrics on a given topological surface is measured by the $L^2$ norm of the deformation tensors. Also mentioned is the several convex geometries naturally occurring in the investigation of Teichmuller space, including the geometries of the Teichmuller metric and the Thurston metric.

Antonio de Nicola, CMUC, Coimbra
Geometry and topology of 3-quasi-Sasakian manifolds

After recalling the main geometric properties of 3-quasi-Sasakian manifolds, I will present our preliminary results on the topology of this class of Riemannian manifolds. They include both 3-cosymplectic and 3-Sasakian manifolds as special cases of minimum and maximum rank, respectively. Recently, there has been an increasing interest toward these manifolds, especially the 3-Sasakian case, due to their relevance in theoretical physics. It is known that any 3-quasi-Sasakian manifold is locally the product of a Hyper-Kaehler manifold and a 3-Sasakian manifold. Nevertheless, we find examples of compact 3-quasi-Sasakian manifolds which are not the global product of a Hyper-Kaehler manifold and a 3-Sasakian manifold.

Richard Gonzales, Institut des Hautes Études Scientifiques (IHES)
Equivariant cohomology of spherical varieties

Let $G$ be a connected reductive group. A $G$-variety is called spherical if it is normal and has a dense orbit under some Borel subgroup of $G$. Examples include flag varieties, symmetric varieties, toric varieties and, more generally, group embeddings. The latter are closely related to algebraic monoids. An important topological property of spherical varieties is that they have only a finite number of $G$-orbits and, consequently, also a finite number of $T$-fixed points, for the induced action of a maximal torus $T$ of $G$. Furthermore, under certain assumptions on the singularity type and the number of $T$-invariant curves, the equivariant cohomology of spherical varieties can be understood via GKM-theory, a technique named after Goresky-Kottwitz-MacPherson. In this talk, we give an overview of these notions and, based on previous work by the author, provide a complete description of the equivariant cohomology of rationally smooth group embeddings. Finally, we report on current work extending these results to the study of equivariant Chow rings and K-theory.

Michael McQuillan, Institut des Hautes Études Scientifiques (IHES)
Hyperbolicity of Algebraic Surfaces

I propose to discuss the main conjecture (the Green-Griffiths conjecture) on the complex hyperbolicity of algebraic surfaces, its consequences in the spirit of the dictum of Bloch: “nihil est in infinito quod non fuerit prius in finito”, and how one proves such things. Various further details may be found at http://www.mat.uniroma2.it/~mcquilla/

Beatriz Graña, Universidad de Salamanca
Jordan-Hölder reductions for principal Higgs bundles on curves

It is known that semistable sheaves V admit a filtration whose quotients are stable and have the same slope of V, named the Jordan-Hölder filtrations. We give the analogous result for principal Higgs bundles on curves.

Ugo Bruzzo, SISSA
Noether-Lefschetz theory for hypersurfaces in toric 3-folds

We prove a sufficient criterion for a very general hypersurface in simplicial toric 3-fold to have the same Picard number as the ambient variety. We also give some preliminary results about the estimate of the codimension of the locus where the Picard number is bigger.

Yoshihiko Mitsumatsu, Chuo University, Tokyo
Regular Poisson structures on S 5 and Milnor's open book

We have constructed a leafwise symplectic structure for a foliation on the 5 sphere which is associated to the Milnor fibration of a simple elliptic hypersurface singularity in 3 complex variables. The foliation associated with x 3 +y 3 +z 3 =0 is nothing but Lawson’s foliation which was constructed as the 1st codimension one foliation on the 5 sphere in 1970. In this talk we abstract the essence of the construction as a framework to produce a leafwise symplectic foliation or a family of almost contact structure converging to a symplectic foliation from an exact symplectic open book decomposition supporting a contact structure.

As a by-product of the framework, we can show that a similar construction works also for the cusp singularities in 3 variables. If the time allows we discuss about the possibility of further applicability of the framework as well as the impossibility.

Nguyen Tien Zung, Institut de Mathématiques de Toulouse
Torus actions in normalization problems of dynamical systems

All dynamical system, including non-hamiltonian and non-integrable ones, admits a natural intrinsic commutative symmetry group (torus action) which preserves not only the system but also any tensor field which is invariant with respect to the system. I will discuss the role played by these torus actions in the normalization problems, including action-angle variables, Poincare-Birkhoff normalization, and renormalization group method.

Sheila Sandon, Université de Nantes & CRM Montreal
The discriminant metric on the contactomorphism group

I will explain the construction of a bi-invariant metric on the universal cover of the contactomorphism group of any contact manifold, and discuss its relations to other contact rigidity phenomena such as the contact non-squeezing theorem, orderability of contact manifolds, and the existence of translated points for contactomorphisms. This is joint work with Vincent Colin.