arXiv subjects

Sílvia Anjos

Publications and source records attributed to Sílvia Anjos.

8 recordsLinked to original sources

arXiv2026

Embedding more than 8 symplectic balls in $\mathbb{C}\mathrm{P}^2$

We prove that the space of symplectic embeddings of $n\geq 1$ standard balls into the standard complex projective plane $\mathbb{C}\mathrm{P}^2$, normalized so that a line has symplectic area $1$, is homotopy equivalent to the configuration space of $n$ points in $\mathbb{C}\mathrm{P}^2$, provided that the sum of the ball capacities is strictly less than $1$. Our techniques further suggest that, for $n=9$, there are infinitely many homotopy types of spaces of symplectic ball embeddings, depending on the ball capacities. Moreover, for each $n\geq 5$, we exhibit capacities for which the embedding spaces are not simply connected, in contrast with the case $n \leq 4$. As an application, we show that, for $n\geq 9$ equal balls of capacity $c<1/n$, the symplectomorphism group of the blow-up has the homotopy type of the stabilizer of $n$ distinct points in $\mathbb{C}\mathrm{P}^2$.

math.SG↗

arXiv2025

Kinematic formulas in convex geometry for non-compact groups

We generalize classical kinematic formulas for convex bodies in a real vector space $V$ to the setting of non-compact Lie groups admitting a Cartan decomposition. Specifically, let $G$ be a closed linear group with Cartan decomposition $G \cong K \times \exp(\mathfrak{p}_0)$, where $K$ is a maximal compact subgroup acting transitively on the unit sphere. For $K$-invariant continuous valuations on convex bodies, we establish an integral geometric-type formula for $\overline{G} = G \ltimes V$. Key to our approach is the introduction of a Gaussian measure on $\mathfrak{p}_0$, which ensures convergence of the non-compact part of the integral. In the special case $K = O(n)$, we recover a Hadwiger-type formula involving intrinsic volumes, with explicit constants $c_j$ computed via a Weyl integration formula.

math.MG↗

arXiv2024

Embeddings of symplectic balls into the complex projective plane

We investigate spaces of symplectic embeddings of $n\leq 4$ balls into the complex projective plane. We prove that they are homotopy equivalent to explicitly described algebraic subspaces of the configuration spaces of $n$ points. We compute the rational homotopy type of these embedding spaces and their cohomology with rational coefficients. Our approach relies on the comparison of the action of $\mathrm{PGL}(3,\mathbb{C})$ on the configuration space of $n$ ordered points in $\mathbf{CP}^2$ with the action of the symplectomorphism group $\mathrm{Symp}(\mathbf{CP}^2)$ on the space of $n$ embedded symplectic balls.

math.SG↗

arXiv2022

Loops in the fundamental group of $\mathrm{Symp} (\mathbb C\mathbb P^2\#\,5\overline{ \mathbb C\mathbb P}\,\!^2)$ which are not represented by circle actions

We study generators of the fundamental group of the group of symplectomorphisms $\mathrm{Symp}({\mathbb C\mathbb P}^2\#\,5\overline{\mathbb C\mathbb P}\,\!^2, ω)$ for some particular symplectic forms. It was observed by J. Kȩdra that there are many symplectic 4-manifolds $(M, ω)$, where $M$ is neither rational nor ruled, that admit no circle action and $π_1 (\mathrm{Ham} (M,ω))$ is nontrivial. On the other hand, it follows from previous results that the fundamental group of the group $\mathrm{Symp}_h({\mathbb C\mathbb P}^2\#\,k\,\overline{\mathbb C\mathbb P}\,\!^2, ω)$, of symplectomorphisms that act trivially on homology, with $k \leq 4$, is generated by circle actions on the manifold. We show that, for some particular symplectic forms $ω$, the set of all Hamiltonian circle actions generates a proper subgroup in $π_1(\mathrm{Symp}_h({\mathbb C\mathbb P}^2\#\,5\overline{\mathbb C\mathbb P}\,\!^2, ω)).$ Our work depends on Delzant classification of toric symplectic manifolds, Karshon's classification of Hamiltonian $S^1$-spaces and the computation of Seidel elements of some circle actions.

math.SG↗

arXiv2018

The homotopy Lie algebra of symplectomorphism groups of 3-fold blow-ups of $(S^2 \times S^2, σ_{std} \oplus σ_{std}) $

We consider the 3-point blow-up of the manifold $ (S^2 \times S^2, σ\oplus σ)$ where $σ$ is the standard symplectic form which gives area 1 to the sphere $S^2$, and study its group of symplectomorphisms $\rm{Symp} ( S^2 \times S^2 \#\, 3\overline{\mathbb C\mathbb P}\,\!^2, ω)$. So far, the monotone case was studied by J. Evans and he proved that this group is contractible. Moreover, J. Li, T. J. Li and W. Wu showed that the group Symp$_{h}(S^2 \times S^2 \#\, 3\overline{ \mathbb C\mathbb P}\,\!^2,ω) $ of symplectomorphisms that act trivially on homology is always connected and recently they also computed its fundamental group. We describe, in full detail, the rational homotopy Lie algebra of this group. We show that some particular circle actions contained in the symplectomorphism group generate its full topology. More precisely, they give the generators of the homotopy graded Lie algebra of $\rm{Symp} (S^2 \times S^2 \#\, 3\overline{ \mathbb C\mathbb P}\,\!^2, ω)$. Our study depends on Karshon's classification of Hamiltonian circle actions and the inflation technique introduced by Lalonde-McDuff. As an application, we deduce the rank of the homotopy groups of $\rm{Symp}({\mathbb C\mathbb P}^2 \#\, 5\overline{\mathbb C\mathbb P}\,\!^2, \tilde ω)$, in the case of small blow-ups.

math.SG↗

arXiv2017

Non-contractible Hamiltonian loops in the kernel of Seidel's representation

The main purpose of this note is to exhibit a Hamiltonian diffeomorphism loop undetected by the Seidel morphism of certain 2-point blow-ups of $S^2 \times S^2$, exactly one of which being monotone. As side remarks, we show that Seidel's morphism is injective on all Hirzebruch surfaces and discuss how to adapt the monotone example to the Lagrangian setting.

math.SG↗

arXiv2015

Seidel's morphism of toric 4-manifolds

Following McDuff and Tolman's work on toric manifolds [McDT06], we focus on 4-dimensional NEF toric manifolds and we show that even though Seidel's elements consist of infinitely many contributions, they can be expressed by closed formulas. From these formulas, we then deduce the expression of the quantum homology ring of these manifolds as well as their Landau-Ginzburg superpotential. We also give explicit formulas for the Seidel elements in some non-NEF cases. These results are closely related to recent work by Fukaya, Oh, Ohta, and Ono [FOOO11], González and Iritani [GI11], and Chan, Lau, Leung, and Tseng [CLLT12]. The main difference is that in the 4-dimensional case the methods we use are more elementary: they do not rely on open Gromov-Witten invariants nor mirror maps. We only use the definition of Seidel's elements and specific closed Gromov-Witten invariants which we compute via localization. So, unlike Alice, the computations contained in this paper are not particularly pretty but they do stay on their side of the mirror. This makes the resulting formulas directly readable from the moment polytope.

math.SG↗

arXiv2012

The homotopy Lie algebra of symplectomorphism groups of 3-fold blow-ups of the projective plane

By a result of Kedra and Pinsonnault, we know that the topology of groups of symplectomorphisms of symplectic 4-manifolds is complicated in general. However, in all known (very specific) examples, the rational cohomology rings of symplectomorphism groups are finitely generated. In this paper, we compute the rational homotopy Lie algebra of symplectomorphism groups of the 3-point blow-up of the projective plane (with an arbitrary symplectic form) and show that in some cases, depending on the sizes of the blow-ups, it is infinite dimensional. Moreover, we explain how the topology is generated by the toric structures one can put on the manifold. Our method involve the study of the space of almost complex structures compatible with the symplectic structure and it depends on the inflation technique of Lalonde-McDuff.

math.SG↗