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Martin Pinsonnault

Publications and source records attributed to Martin Pinsonnault.

17 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 $\mathbb{C}\mathrm{P}^2$.

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arXiv2025

Centralizers of Hamiltonian finite cyclic group actions on rational ruled surfaces

Let $M=(M,ω)$ be either the product $S^2\times S^2$ or the non-trivial $S^2$ bundle over $S^2$ endowed with any symplectic form $ω$. Suppose a finite cyclic group $Z_n$ is acting effectively on $(M,ω)$ through Hamiltonian diffeomorphisms, that is, there is an injective homomorphism $Z_n\hookrightarrow Ham(M,ω)$. In this paper, we investigate the homotopy type of the group $Symp^{Z_n}(M,ω)$ of equivariant symplectomorphisms. We prove that for some infinite families of $Z_n$ actions satisfying certain inequalities involving the order $n$ and the symplectic cohomology class $[ω]$, the actions extends to either one or two toric actions, and accordingly, that the centralizers are homotopically equivalent to either a finite dimensional Lie group, or to the homotopy pushout of two tori along a circle. Our results rely on $J$-holomorphic techniques, on Delzant's classification of toric actions, on Karshon's classification of Hamiltonian circle actions on $4$-manifolds, and on the Chen-Wilczyński classification of smooth $Z_n$-actions on Hirzebruch surfaces.

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arXiv2025

Centralizers of Hamiltonian circle actions on rational ruled surfaces

In this paper, we compute the homotopy type of the group of equivariant symplectomorphisms of $S^2 \times S^2$ and $\mathbb{C}P^2 \# \overline{\mathbb{C}P^2}$ 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. This follows from the analysis of the action of equivariant symplectomorphisms on the space of compatible and invariant almost complex structures $\mathcal{J}^{S^1}_ω$. In particular, we show that this action preserves a decomposition of $\mathcal{J}^{S^1}_ω$ into strata which are in bijection with toric extensions of the circle action. Our results rely on $J$-holomorphic techniques, on Delzant's classification of toric actions and on Karshon's classification of Hamiltonian circle actions on $4$-manifolds.

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arXiv2025

J-tamed inflation via tame to compatible deformations

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 $(M,ω)$, and given any compact, embedded, $J$-holomorphic submanifold $Z$, it is always possible to construct a deformation of symplectic forms $ω_t$ in classes $[ω_t]=[ω]+t\mathrm{PD}{Z}$, for $0\leq t$ less than an upper bound $0<T$ that only depends on the self-intersection $Z\cdot Z$. The original proofs of this fact make the unwarranted assumption that one can find a family of normal planes along $Z$ that is both $J$ invariant and $ω$-orthogonal to $TZ$ -- which amounts, in effect, to assuming the compatibility of $J$ and $ω$ along $Z$. We explain how the original constructions can be adapted to avoid this assumption when $Z$ has nonpositive self-intersection, and we discuss the difficulties with this line of argument in general to establish the full inflation when $Z$ has positive self-intersection. We overcome this problem by proving a `preparation lemma', which states that prior to inflation, one can isotope $ω$ within its cohomology class to a new form that still tames $J$ and which is compatible with $J$ along the submanifold $Z$. This preparation lemma can be regarded as an infinitesimal version of the "tamed-to-compatible" conjecture of S. K. Donaldson along an almost-complex submanifold $Z$.

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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.

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arXiv2023

Stability of the symplectomorphism group of rational surfaces

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 $12$. As a corollary, we also derive a stability result for the space of embedded symplectic balls in $M$. A noteworthy feature of our approach is that we systematically explore various spaces and groups associated to a symplectic cohomology class $u$ rather than with a single symplectic form $ω$. To this end, we prove a weaker version of the tamed $J$-inflation procedures of D. McDuff and O. Buse that fixes a gap in their original formulations.

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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.

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arXiv2014

Counting toric actions on symplectic four-manifolds

Given a symplectic manifold, we ask in how many different ways can a torus act on it. Classification theorems in equivariant symplectic geometry can sometimes tell that two Hamiltonian torus actions are inequivalent, but often they do not tell whether the underlying symplectic manifolds are (non-equivariantly) symplectomorphic. For two dimensional torus actions on closed symplectic four-manifolds, we reduce the counting question to combinatorics, by expressing the manifold as a symplectic blowup in a way that is compatible with all the torus actions simultaneously. For this we use the theory of pseudoholomorphic curves.

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arXiv2013

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

We establish connections between contact isometry groups of certain contact manifolds and compactly supported symplectomorphism groups of their symplectizations. We apply these results to investigate the space of symplectic embeddings of balls with a single conical singularity at the origin. Using similar ideas, we also prove the longstanding expected result that the space of Lagrangian $\RR P^2$ in $T^*\RR P^2$ is weakly contractible.

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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.

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arXiv2011

Packing numbers of rational ruled 4-manifolds

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. As a corollary, we give explicit values for when an ellipsoid of type $E(a, b)$, with $\frac{b}{a} \in \N$, embeds in a polydisc $P(s,t)$. Under this integrality assumption, we also give an alternative proof of a recent result of M. Hutchings showing that the ECH capacities give sharp inequalities for embedding ellipsoids into polydisks.

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arXiv2009

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

Let $X$ be any rational ruled symplectic four-manifold. Given a symplectic embedding $ι:B_{c}\into X$ of the standard ball of capacity $c$ into $X$, consider the corresponding symplectic blow-up $\tX_ι$. In this paper, we study the homotopy type of the symplectomorphism group $\Symp(\tX_ι)$, simplifying and extending the results of math.SG/0207096. This allows us to compute the rational homotopy groups of the space $\IEmb(B_{c},X)$ of unparametrized symplectic embeddings of $B_{c}$ into $X$. We also show that the embedding space of one ball in $CP^2$, and the embedding space of two disjoint balls in $CP^2$, if non empty, are always homotopy equivalent to the corresponding spaces of ordered configurations. Our method relies on the theory of pseudo-holomorphic curves in 4-manifolds, on the theory of Gromov invariants, and on the inflation technique of Lalonde-McDuff.

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arXiv2008

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

Let M:=(M^{4},\om) be a 4-dimensional rational ruled symplectic manifold and denote by w_{M} its Gromov width. Let Emb_ω(B^{4}(c),M) be the space of symplectic embeddings of the standard ball B^4(c) \subset \R^4 of radius r and of capacity c:= πr^2 into (M,\om). By the work of Lalonde and Pinsonnault, we know that there exists a critical capacity \ccrit \in (0,w_{M}] such that, for all c\in(0,\ccrit), the embedding space Emb_ω(B^{4}(c),M) is homotopy equivalent to the space of symplectic frames \SFr(M). We also know that the homotopy type of Emb_ω(B^{4}(c),M) changes when c reaches \ccrit and that it remains constant for all c \in [\ccrit,w_{M}). In this paper, we compute the rational homotopy type, the minimal model, and the cohomology with rational coefficients of \Emb_ω(B^{4}(c),M) in the remaining case c \in [\ccrit,w_{M}). In particular, we show that it does not have the homotopy type of a finite CW-complex.

math.SG↗
arXiv2006

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

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. We also extend to rational and ruled manifolds a result of Kedra which asserts that, if $M$ is a simply connected symplectic 4-manifold with $b_{2}\geq 3$, and if $\widetilde{M}_δ$ denotes a blow-up of $M$ of small enough capacity $δ$, then the rational cohomology algebra of the Hamiltonian group of $\widetilde{M}_δ)$ is not finitely generated. Both results are based on the fact that in a symplectic 4-manifold endowed with any tamed almost structure $J$, exceptional classes of minimal symplectic area are $J$-indecomposable. Some applications and examples are given.

math.SG↗
arXiv2006

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

Let (M,ω) be a four dimensional compact connected symplectic manifold. We prove that (M,ω) admits only finitely many inequivalent Hamiltonian effective 2-torus actions. Consequently, if M is simply connected, the number of conjugacy classes of 2-tori in the symplectomorphism group Sympl(M,ω) is finite. Our proof is "soft". The proof uses the fact that for symplectic blow-ups of \CP^2 the restriction of the period map to the set of exceptional homology classes is proper. In an appendix, we describe results of McDuff that give a properness result for a general compact symplectic four-manifold, using the theory of J-holomorphic curves.

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arXiv2004

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

We study in this paper the rational homotopy type of the space of symplectic embeddings of the standard ball $B^4(c) \subset \R^4$ into 4-dimensional rational symplectic manifolds. We compute the rational homotopy groups of that space when the 4-manifold has the form $M_λ= (S^2 \times S^2, μω_0 \oplus ω_0)$ where $ω_0$ is the area form on the sphere with total area 1 and $μ$ belongs to the interval $[1,2]$. We show that, when $μ$ is 1, this space retracts to the space of symplectic frames, for any value of $c$. However, for any given $1 < μ< 2$, the rational homotopy type of that space changes as $c$ crosses the critical parameter $c_{crit} = μ- 1$, which is the difference of areas between the two $S^2$ factors. We prove moreover that the full homotopy type of that space changes only at that value, i.e the restriction map between these spaces is a homotopy equivalence as long as these values of $c$ remain either below or above that critical value.

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