Kähler and special toric geometry

Schedule

The meeting will run from 11am Monday to 4pm Wednesday. Most of the talks will be in Bush House BH(SE)2.09. Use the main entrance at the intersection of the Aldwych with the south end of Kingsway. Registration is essential to gain access. The last talk will be held in UCL, Gordon Street, near to St Pancras Station.

Monday, May 20	Tuesday, May 21	Wednesday, May 22
	10:00 Yann Rollin Discrete geometry and isotropic surfaces	10:00 Nicolina Istrati Toric locally conformally Kähler metrics
10:30 Coffee/tea	11:00 Coffee/tea	11:00 Coffee/tea
11:00 Michael Singer Yet another construction of ALF $D_k$ gravitational instantons	11:30 Eveline Legendre Toric extremal almost Kähler metrics	11:30 Hugues Auvray Complete extremal metrics and stability of pairs on Hirzebruch surfaces
12:00 Lunch	12:30 Lunch	12:30 Lunch
2:00 Kael Dixon Toric G2 and nearly Kaehler geometry	2:30 Dmitri Panov Symplectic Fano manifolds with Hamiltonian $S^1$-actions	3:00 at UCL Drayton House B03 Frances Kirwan Moment maps and non-reductive geometric invariant theory
3:00 Tea/coffee	3:30 Tea/coffee
3:30 Paul Gauduchon Around the Gibbons-Hawking Ansatz	6:30 Dinner at Navarro's 67 Charlotte St, Fitzrovia, London W1T 4PH

Home page

Abstracts

Hugues Auvray: Complete extremal metrics and stability of pairs on Hirzebruch surfaces

In this talk, I will discuss the existence of complete extremal metrics on the complement of simple normal crossings divisors in compact Kähler manifolds, and stability of pairs, in the toric case. Using constructions of Legendre and Apostolov-Calderbank-Gauduchon, we completely characterize when this holds for Hirzebruch surfaces. In particular, our results show that relative stability of a pair and the existence of extremal Poincaré type/cusp metrics do not coincide. However, stability is equivalent to the existence of a complete extremal metric on the complement of the divisor in our examples. It is the Poincaré type condition on the asymptotics of the extremal metric that fails in general. This is joint work with Vestislav Apostolov and Lars Sektnan.

Kael Dixon: Toric G2 and nearly Kaehler geometry

We will discuss analogues of toric geometry in the G2 and nearly Kaehler settings using a multisymplectic generalization of the moment map called the multi-moment map. We will then present recent work on complete toric nearly Kaehler manifolds, demonstrating some information about their global structure and giving evidence to the conjecture that the homogeneous nearly Kaehler structure on the product of two three spheres is the only example.

Paul Gauduchon: Around the Gibbons-Hawking Ansatz

Roughly speaking, the Gibbons-Hawking Ansatz associates to a positive harmonic function $V$ defined on some open subset of $\mathbb{R} ^3$ a $S ^1$-invariant, four-dimensional structure. Unless $V$ is constant, to any such metric is canonically associated an almost Kaehler structure defined in the same conformal class and inducing the oppositie orientation. We show that the latter is Kaehler if and only if $V$ is either affine or a radial function. In the Taub-NUT case, the resulting ambitoric structure turns out to be "of parabolic type" and we identify the "negative partner" with one of the complete self-dual Kähler strucures constructed by Robert Bryant on the complex plane.

Nicolina Istrati: Toric locally conformally Kähler metrics

Locally conformally Kahler (LCK) metrics are generalizations of Kahler metrics in a conformal manner. Toric geometry adapts naturally to this context. I will give an introduction to toric LCK manifolds, and then I will show that any toric LCK metric on a compact manifold admits a positive potential.

Frances Kirwan: Moment maps and non-reductive geometric invariant theory

When a complex reductive group acts linearly on a projective variety the quotient in the sense of geometric invariant theory (GIT) can be identified with an appropriate symplectic quotient. The aim of this talk is to discuss an analogue of this description for GIT quotients by suitable non-reductive actions. In general GIT for non-reductive linear algebraic group actions is much less well behaved than for reductive actions. However when a linear algebraic group has internally graded unipotent radical U, in the sense that a Levi subgroup has a central one-parameter subgroup which acts by conjugation on U with all weights strictly positive, then GIT for a linear action of the group on a projective variety is almost as well behaved as in the reductive setting, provided that we are willing to multiply the linearisation by an appropriate rational character. In this situation we can ask for a moment map description of the quotient. This is related to the symplectic implosion construction (introduced in a 2002 paper of Guillemin, Jeffrey and Sjamaar) and recent work by Greb and Miebach on Hamiltonian actions of unipotent groups on compact Kähler manifolds.

Eveline Legendre: Toric extremal almost Kähler metrics

I will show how some results of Chen--Li--Sheng and He concerning extremal Kähler metrics and uniform K-stability hold for toric extremal almost Kähler metrics. As a consequence, we get that a toric symplectic compact manifold admits a compatible extremal Kähler metric if and only if it admits a compatible extremal almost Kähler metric.

Dmitri Panov: Symplectic Fano manifolds with Hamiltonian $S^1$-actions

A compact symplectic manifold (M,w) is called Fano if the classes $c_1(M)$ and [w] coincide in $H^2(M)$. After discussing low-dimensional examples and a construction of non-algebraic symplectic Fanos in dimension 12 and higher (joined with Joel Fine) I will talk about my work with Nick Lindsay. We prove that any symplectic Fano 6-manifold M with a Hamiltonian $S^1$-action is simply connected and satisfies $c_1c_2(M)=24$. This is done by showing that the fixed submanifold of M on which the Hamiltonian attains its minimum is diffeomorphic to either a del Pezzo surface, a 2-sphere or a point.

Yann Rollin: Discrete geometry and isotropic surfaces

We consider smooth isotropic immersions from the 2-dimensional torus into $\mathbb R^{2n}$, for n≥2. When n=2 the image of such map is an immersed Lagrangian torus of $\mathbb R^4$. We prove that such isotropic immersions can be approximated by arbitrarily $C_0$-close piecewise linear isotropic maps. If n≥3 the piecewise linear isotropic maps can be chosen so that they are piecewise linear isotropic immersions as well. The proofs are obtained using analogies with an infinite dimensional moment map geometry due to Donaldson. As a byproduct of these considerations, we introduce a numerical flow in finite dimension, whose limit provide, from an experimental perspective, many examples of piecewise linear Lagrangian tori in $\mathbb R^4$.