Publications

Remarks on $J$-tame inflation

Published in arXiv.org, 2024

We give a complete and self-contained exposition of the $J$-tame inflation lemma: Given any tame almost complex structure $J$ on a symplectic 4-manifold $(M,\omega)$, and given any compact, embedded, $J$-holomorphic submanifold $Z$, it is always possible to construct a deformation of symplectic forms $\omega_t$ in classes $[\omega_t]=[\omega]+tPD(Z)$, for $0\leq t$ less than an upper bound $0< T$ that only depends on the self-intersection $Z\cdot Z$.

Recommended citation: Pranav Chakravarthy, Jordan Payette, Martin Pinsonnault. (2024). "Remarks on $J$-tame inflation." arXiv:2403.19110. https://arxiv.org/abs/2403.19110

Embeddings of symplectic balls into the complex projective plane

Published in arxiv.org, 2024

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.

Recommended citation: Sílvia Anjos, Jarek Kędra, Martin Pinsonnault. (2024). "Embeddings of symplectic balls into the complex projective plane." arxiv:2307.00556. https://arxiv.org/abs/2307.00556

Centralizers of Hamiltonian finite cyclic group actions on rational ruled surfaces

Published in Transactions of the AMS, 2023

Let $M=(M,\omega)$ be either the product $S^2\times S^2$ or the non-trivial $S^2$ bundle over $S^2$ endowed with any symplectic form $\omega$. Suppose a finite cyclic group $\mathbf{Z}_n$ is acting effectively on $(M,\omega)$ through Hamiltonian diffeomorphisms, that is, there is an injective homomorphism $\mathbf{Z}_n\hookrightarrow \mathrm{Ham}(M,\omega)$. In this paper, we investigate the homotopy type of the group $\mathrm{Symp}^{\mathbf{Z}_n}(M,\omega)$ of equivariant symplectomorphisms.

Recommended citation: Pranav V. Chakravarthy, Martin Pinsonnault. (2023). "Centralizers of Hamiltonian finite cyclic group actions on rational ruled surfaces." https://doi.org/10.1090/tran/9223. https://doi.org/10.1090/tran/9223

Stability of the symplectomorphism groups of rational surfaces

Published in Mathematische Annalen, 2023

We apply Zhang’s almost Kähler Nakai–Moishezon theorem and Li–Zhang’s comparison of J-symplectic cones to establish a stability result for the symplectomorphism group of a rational 4-manifold $M$ with Euler number up to 12.

Recommended citation: Anjos, S., Li, J., Li, TJ. et al. Stability of the symplectomorphism groups of rational surfaces. Math. Ann. 389, 85–119 (2024). https://doi-org.proxy1.lib.uwo.ca/10.1007/s00208-023-02643-5 https://doi.org/10.1007/s00208-023-02643-5

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

Published in Canadian Journal of Mathematics. Journal Canadien de Mathématiques, 2023

We study generators of the fundamental group of the group of symplectomorphisms $\mathrm{Symp}(\mathbf{C}P^2\sharp 5\overline{\mathbf{C}P^2},\omega)$ for some particular symplectic forms.

Recommended citation: Anjos, S., Barata, M., Pinsonnault, M., Reis, A. Loops in the fundamental group of $\mathbf{Symp}(\overline{\mathbf{C}P^2\sharp 5\mathbf{C}P^2},\omega)$ which are not represented by circle actions. Canad. J. Math.75 (2023), no.4, 1226–1271. https://doi.org/10.4153/s0008414x22000323

Centralizers of Hamiltonian circle actions on rational ruled surfaces

Published in arxiv.org, 2022

In this paper, we compute the homotopy type of the group of equivariant symplectomorphisms of $S^2\times S^2$ and $\mathbf{C}P^2\sharp\overline{\mathbf{C}P^2}$ under the presence of Hamiltonian group actions of the circle $S^1$. We prove that the group of equivariant symplectomorphisms are homotopy equivalent to either a torus, or to the homotopy pushout of two tori depending on whether the circle action extends to a single toric action or to exactly two non-equivalent toric actions.

Recommended citation: Pranav V. Chakravarthy, Martin Pinsonnault. (2023). "Centralizers of Hamiltonian circle actions on rational ruled surfaces." arxiv:2202.08255 . https://arxiv.org/abs/2202.08255

Symplectormophism groups of non-compact manifolds, orbifold balls, and a space of Lagrangians

Published in Journal of Symplectic Geometry, 2016

We establish connections between contact isometry groups of certain contact manifolds and compactly supported symplectomorphism groups of their symplectizations.

Recommended citation: Hind, R., Pinsonnault, M., and Wu, W. (2024). "Symplectormophism groups of non-compact manifolds, orbifold balls, and a space of Lagrangians." J. Symplectic Geom.14 (2016), no.1, 203–226. https://doi.org/10.4310/JSG.2016.v14.n1.a8

Counting toric actions on symplectic four-manifolds

Published in C. R. Math. Acad. Sci. Soc. R. Can., 2014

Given a symplectic manifold, we ask in how many different ways can a torus act on it.

Recommended citation: Karshon, Y., Kessler, L., Pinsonnault, M. "Counting toric actions on symplectic four-manifolds." C. R. Math. Acad. Sci. Soc. R. Can. 37 (2015), no.1, 33–40. https://arxiv.org/abs/1409.6061

Semi-classical properties of Berezin-Toeplitz operators with $\mathcal{C}^k$-symbol

Published in J. Math. Phys., 2014

We obtain the semi-classical expansion of the kernels and traces of Toeplitz operators with $\mathcal{C}^k$-symbol on a symplectic manifold. We also give a semi-classical estimate of the distance of a Toeplitz operator to the space of self-adjoint and multiplication operators.

Recommended citation: Barron, T., Ma, X., Marinescu, G., Pinsonnault, M. "Semi-classical properties of Berezin-Toeplitz operators with $\mathcal{C}^k$-symbol." J. Math. Phys.55 (2014), no.4, 042108, 25 pp. https://doi-org.proxy1.lib.uwo.ca/10.1063/1.4870869

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

Published in Mathematische Zeitschrift, 2012

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.

Recommended citation: Anjos, S., Pinsonnault, M., The homotopy Lie algebra of symplectomorphism groups of 3-fold blow-ups of the projective plane. Math. Z. 275 (2013), no.1-2, 245–292. https://doi.org/10.1007/s00209-012-1134-5

Packing numbers of rational ruled four-manifolds

Published in The Journal of Symplectic Geometry, 2011

We completely solve the symplectic packing problem with equally sized balls for any rational, ruled, symplectic 4-manifolds. We give explicit formulae for the packing numbers, the generalized Gromov widths, the stability numbers, and the corresponding obstructing exceptional classes.

Recommended citation: Buse, O., Pinsonnault, M., Packing numbers of rational ruled four-manifolds. J. Symplectic Geom. 11 (2013), no.2, 269–316. https://doi.org/10.4310/JSG.2013.v11.n2.a5

The homotopy type of the space of symplectic balls in rational ruled 4-manifolds

Published in Geometry & Topology, 2008

In this paper, we compute the rational homotopy type, the minimal model, and the cohomology with rational coefficients of the embedding space $\mathrm{Emb}_{\omega}(B^{4}(c),M)$ for a “large” ball of capacity $c \in [c_{\mathrm{crit}},w_{M})$. In particular, we show that it does not have the homotopy type of a finite CW-complex.

Recommended citation: Anjos, S., Lalonde, F., Pinsonnault, M., The homotopy type of the space of symplectic balls in rational ruled 4-manifolds. Geom. Topol. 13 (2009), no.2, 1177–1227. https://doi.org/10.2140/gt.2009.13.1177

Maximal compact tori in the Hamiltonian groups of 4-dimensional symplectic manifolds

Published in Journal of Modern Dynamics, 2006

We prove that the group of Hamiltonian automorphisms of a symplectic 4-manifold contains only finitely many conjugacy classes of maximal compact tori with respect to the action of the full symplectomorphism group.

Recommended citation: Pinsonnault, M., Maximal compact tori in the Hamiltonian group of 4-dimensional symplectic manifolds. J. Mod. Dyn. 2 (2008), no.3, 431–455. https://doi.org/10.3934/jmd.2008.2.431

A compact symplectic four-manifold admits only finitely many inequivalent toric actions

Published in The Journal of Symplectic Geometry, 2006

Let $(M,\omega)$ be a four dimensional compact connected symplectic manifold. We prove that $(M,\omega)$ admits only finitely many inequivalent Hamiltonian effective 2-torus actions.

Recommended citation: Karshon, Y., Kessler, L., Pinsonnault, M., A compact symplectic four-manifold admits only finitely many inequivalent toric actions. J. Symplectic Geom. 5 (2007), no.2, 139–166. http://projecteuclid.org/euclid.jsg/1202004454

Symplectomorphism groups and embeddings of balls into rational ruled 4-manifolds

Published in Compositio Mathematica, 2006

In this paper, we study the homotopy type of the symplectomorphism group $\mathrm{Symp}(\tilde X_\iota)$, simplifying and extending the results of math.SG/0207096. This allows us to compute the rational homotopy groups of the space $\Im\mathrm{Emb}(B_{c},X)$ of unparametrized symplectic embeddings of $B_{c}$ into $X$.

Recommended citation: Pinsonnault, M., Symplectomorphism groups and embeddings of balls into rational ruled 4-manifolds. Compos. Math. 144 (2008), no.3, 787–810. https://doi.org/10.1112/S0010437X0700334X

The topology of the space of symplectic balls in rational 4-manifolds

Published in Duke Mathematical Journal, 2004

We study in this paper the rational homotopy type of the space of symplectic embeddings of the standard ball $B^4(c) \subset \mathbf{R}^4$ into 4-dimensional rational symplectic manifolds. We compute the rational homotopy groups of that space when the 4-manifold has the form $M_{\mu}=(S^2\times S^2,\mu\omega_0\oplus\omega_0)$ where $\omega_0$ is the area form on the sphere with total area 1 and $\mu$ belongs to the interval $[1,2]$.

Recommended citation: Lalonde, F., Pinsonnault, M., The topology of the space of symplectic balls in rational 4-manifolds. Duke Math. J. 122 (2004), no.2, 347–397. https://doi.org/10.1215/S0012-7094-04-12223-7

Groupes d' automorphismes et plongements symplectiques de boules dans les variétés rationnelles

Published in Comptes Rendus Mathématique de l' Académie des Sciences de Paris, 2002

Nous étudions l’espace $Pl(c,\lambda)$ des plongements symplectiques de la boule fermée $B^4(c) \subset\mathbf{R}^4$ de capacité $c$ dans $(S^2 \times S^2,(1+\lambda)\omega_{st}\oplus\omega_{st})$.

Recommended citation: Lalonde, F., Pinsonnault, M., Groupes d' automorphismes et plongements symplectiques de boules dans les variétés rationnelles. C. R. Math. Acad. Sci. Paris 335 (2002), no.11, 931–934. https://doi.org/10.1016/S1631-073X(02)02583-9