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Miguel Abreu

Publications and source records attributed to Miguel Abreu.

31 records · Page 2Linked to original sources
arXiv2012

Remarks on Lagrangian intersections in toric manifolds

We consider two natural Lagrangian intersection problems in the context of symplectic toric manifolds: displaceability of torus orbits and of a torus orbit with the real part of the toric manifold. Our remarks address the fact that one can use simple cartesian product and symplectic reduction considerations to go from basic examples to much more sophisticated ones. We show in particular how rigidity results for the above Lagrangian intersection problems in weighted projective spaces can be combined with these considerations to prove analogous results for all monotone toric symplectic manifolds. We also discuss non-monotone and/or non-Fano examples, including some with a continuum of non-displaceable torus orbits.

math.SG↗
arXiv2011

Contact homology of good toric contact manifolds

In this paper we show that any good toric contact manifold has well defined cylindrical contact homology and describe how it can be combinatorially computed from the associated moment cone. As an application we compute the cylindrical contact homology of a particularly nice family of examples that appear in the work of Gauntlett-Martelli-Sparks-Waldram on Sasaki-Einstein metrics. We show in particular that these give rise to a new infinite family of non-equivalent contact structures on $S^2 \times S^{3}$ in the unique homotopy class of almost contact structures with vanishing first Chern class.

math.SG↗
arXiv2011

Scalar-flat Kahler metrics on non-compact symplectic toric 4-manifolds

In a recent paper Donaldson explains how to use an older construction of Joyce to obtain four dimensional local models for scalar-flat Kahler metrics with a 2-torus symmetry. Using this idea, he recovers and generalizes the Taub-NUT metric by including it in a new family of complete scalar-flat toric Kahler metrics. In this paper we generalize Donaldson's method and construct complete scalar-flat toric Kahler metrics on any symplectic toric 4-manifold with "strictly unbounded" moment polygon. These include the asymptotic locally Euclidean scalar-flat Kahler metrics previously constructed by Calderbank and Singer, as well as new examples of complete scalar-flat toric Kahler metrics which are asymptotic to Donaldson's generalized Taub-NUT metrics. Our construction is in symplectic action-angle coordinates and determines all these metrics via their symplectic potentials. When the first Chern class is zero we obtain a new description of known Ricci-flat Kahler metrics.

math.DG↗
arXiv2010

Kahler-Sasaki geometry of toric symplectic cones in action-angle coordinates

In the same way that a contact manifold determines and is determined by a symplectic cone, a Sasaki manifold determines and is determined by a suitable Kahler cone. Kahler-Sasaki geometry is the geometry of these cones. This paper presents a symplectic action-angle coordinates approach to toric Kahler geometry and how it was recently generalized, by Burns-Guillemin-Lerman and Martelli-Sparks-Yau, to toric Kahler-Sasaki geometry. It also describes, as an application, how this approach can be used to relate a recent new family of Sasaki-Einstein metrics constructed by Gauntlett-Martelli-Sparks-Waldram in 2004, to an old family of extremal Kahler metrics constructed by Calabi in 1982.

math.DG↗
arXiv2009

Compatible complex structures on symplectic rational ruled surfaces

In this paper we study the topology of the space $\I_ω$ of complex structures compatible with a fixed symplectic form $ω$, using the framework of Donaldson. By comparing our analysis of the space $\I_ω$ with results of McDuff on the space $\cat J_ω$ of compatible almost complex structures on rational ruled surfaces, we find that $\I_ω$ is contractible in this case. We then apply this result to study the topology of the symplectomorphism group of a rational ruled surface, extending results of Abreu and McDuff.

math.SG↗
arXiv2006

Hearing the weights of weighted projective planes

Which properties of an orbifold can we ``hear,'' i.e., which topological and geometric properties of an orbifold are determined by its Laplace spectrum? We consider this question for a class of four-dimensional Kähler orbifolds: weighted projective planes $M:=\C P^2(N_1,N_2,N_3)$ with three isolated singularities. We show that the spectra of the Laplacian acting on 0- and 1-forms on $M$ determine the weights $N_1$, $N_2$, and $N_3$. The proof involves analysis of the heat invariants using several techniques, including localization in equivariant cohomology. We show that we can replace knowledge of the spectrum on 1-forms by knowledge of the Euler characteristic and obtain the same result. Finally, after determining the values of $N_1$, $N_2$, and $N_3$, we can hear whether $M$ is endowed with an extremal Kähler metric.

math.DG↗
arXiv2005

Moment maps, symplectomorphism groups and compatible complex structures

In this paper we apply Donaldson's general moment map framework for the action of a symplectomorphism group on the corresponding space of compatible (almost) complex structures to the case of rational ruled surfaces. This gives a new approach to understanding the topology of their symplectomorphism groups, based on a result of independent interest: the space of compatible integrable complex structures on any symplectic rational ruled surface is (weakly) contractible. We also explain how in general, under this condition, there is a direct relationship between the topology of a symplectomorphism group, the deformation theory of compatible complex structures and the groups of complex automorphisms of these complex structures.

math.SG↗
arXiv2001

On the invariant spectrum of $S^1$-invariant metrics on $S^2$

A theorem of J. Hersch (1970) states that for any smooth metric on $S^2$, with total area equal to $4π$, the first nonzero eigenvalue of the Laplace operator acting on functions is less than or equal to 2 (this being the value for the standard round metric). For metrics invariant under the standard $S^1$-action on $S^2$, one can restrict the Laplace operator to the subspace of $S^1$-invariant functions and consider its spectrum there. The corresponding eigenvalues will be called invariant eigenvalues, and the purpose of this paper is to analyse its possible values. We first show that there is no general analogue of Hersch's theorem, by exhibiting explicit families of $S^1$-invariant metrics with total area $4π$ where the first invariant eigenvalue ranges through any value between 0 and $\infty$. We then restrict ourselves to $S^1$-invariant metrics that can be embedded in ${\bf R}^3$ as surfaces of revolution. For this subclass we are able to provide optimal upper bounds for all invariant eigenvalues. As a consequence, we obtain an analogue of Hersch's theorem with an optimal upper bound (greater than 2 and geometrically interesting). This subclass of metrics on $S^2$ includes all $S^1$-invariant metrics with non-negative Gauss curvature. One of the key ideas in the proofs of these results comes from symplectic geometry, and amounts to the use of the moment map of the $S^1$-action as a coordinate function on $S^2$.

math.SP↗
arXiv2001

Kahler metrics on toric orbifolds

A theorem of E.Lerman and S.Tolman, generalizing a result of T.Delzant, states that compact symplectic toric orbifolds are classified by their moment polytopes, together with a positive integer label attached to each of their facets. In this paper we use this result, and the existence of "global" action-angle coordinates, to give an effective parametrization of all compatible toric complex structures on a compact symplectic toric orbifold, by means of smooth functions on the corresponding moment polytope. This is equivalent to parametrizing all toric Kahler metrics and generalizes an analogous result for toric manifolds. A simple explicit description of interesting families of extremal Kahler metrics, arising from recent work of R.Bryant, is given as an application of the approach in this paper. The fact that in dimension four these metrics are self-dual and conformally Einstein is also discussed. This gives rise in particular to a one parameter family of self-dual Einstein metrics connecting the well known Eguchi-Hanson and Taub-NUT metrics.

math.DG↗
arXiv2000

Kahler geometry of toric manifolds in symplectic coordinates

A theorem of Delzant states that any symplectic manifold $(M,\om)$ of dimension $2n$, equipped with an effective Hamiltonian action of the standard $n$-torus $\T^n = \R^{n}/2π\Z^n$, is a smooth projective toric variety completely determined (as a Hamiltonian $\T^n$-space) by the image of the moment map $ϕ:M\to\R^n$, a convex polytope $P=ϕ(M)\subset\R^n$. In this paper we show, using symplectic (action-angle) coordinates on $P\times \T^n$, how all $\om$-compatible toric complex structures on $M$ can be effectively parametrized by smooth functions on $P$. We also discuss some topics suited for application of this symplectic coordinates approach to Kähler toric geometry, namely: explicit construction of extremal Kähler metrics, spectral properties of toric manifolds and combinatorics of polytopes.

math.DG↗
arXiv1999

Topology of symplectomorphism groups of rational ruled surfaces

Let $M$ be either $S^2\times S^2$ or the one point blow-up $\cp# \bcp$ of $\cp$. In both cases $M$ carries a family of symplectic forms $\om_\la$, where $\la > -1$ determines the cohomology class $[\om_\la]$. This paper calculates the rational (co)homology of the group $G_\la$ of symplectomorphisms of $(M,\om_\la)$ as well as the rational homotopy type of its classifying space $BG_\la$. It turns out that each group $G_\la$ contains a finite collection $K_k, k = 0,...,\ell = \ell(\la)$, of finite dimensional Lie subgroups that generate its homotopy. We show that these subgroups "asymptotically commute", i.e. all the higher Whitehead products that they generate vanish as $\la\to \infty$. However, for each fixed $\la$ there is essentially one nonvanishing product that gives rise to a "jumping generator" $w_\la$ in $H^*(G_\la)$ and to a single relation in the rational cohomology ring $H^*(BG_\la)$. An analog of this generator $w_\la$ was also seen by Kronheimer in his study of families of symplectic forms on 4-manifolds using Seiberg--Witten theory. Our methods involve a close study of the space of $\om_\la$-compatible almost complex structures on $M$.

math.SG↗
arXiv1997

Kahler geometry of toric varieties and extremal metrics

Recently Guillemin gave an explicit combinatorial way of constructing "toric" Kahler metrics on (symplectic) toric varieties, using only data on the moment polytope. In this paper, differential geometric properties of these metrics are investigated using Guillemin's construction. In particular, a nice combinatorial formula for the scalar curvature is given, and the Euler-Lagrange condition for such "toric" metrics being extremal (in the sense of Calabi) is derived. A construction, due to Calabi, of a 1-parameter family of extremal metrics of non-constant scalar curvature is recast very simply and explicitly. Finally, a curious combinatorial formula for convex polytopes, that follows from the relation between the total integral of the scalar curvature and the wedge product of the first Chern class with a suitable power of the Kahler class, is presented.

dg-ga↗