arXiv subjects

Isabelle Charton

Publications and source records attributed to Isabelle Charton.

5 recordsLinked to original sources

arXiv2026

K\"ahler complexity one Hamiltonian $T$-manifolds have trivial paintings

Let a torus $T$ act on a symplectic manifold $(M,\omega)$ with moment map $\phi$. We say that the Hamiltonian $T$-manifold $(M,\omega,\phi)$ has complexity one if $\frac{1}{2} \dim M - \dim T = 1$, and that it is K\"ahler if it admits an invariant compatible complex structure. In this paper, we show how the class of K\"ahler complexity one Hamiltonian $T$-manifolds sits inside the class of complexity one Hamiltonian $T$-manifolds by proving that every compact, connected K\"ahler complexity one Hamiltonian $T$-manifold has a trivial painting. As a corollary, we show that two tall compact, connected K\"ahler complexity one Hamiltonian $T$-manifolds are symplectomorphic exactly if they have the same genus, Duistermaat-Heckman measure, and skeleton. Here, $(M,\omega,\phi)$ is tall exactly if every non-empty fiber $\phi^{-1}(\alpha)$ contains more than one orbit.

math.SG↗

arXiv2023

Monotone Symplectic Six-Manifolds that admit a Hamiltonian GKM Action are diffeomorphic to Smooth Fano Threefolds

Let $(M,ω)$ be a compact symplectic manifold with a Hamiltonian GKM action of a compact torus. We formulate a positive condition on the space; this condition is satisfied if the underlying symplectic manifold is monotone. The main result of this article is that the underlying manifold of a positive Hamiltonian GKM space of dimension six is diffeomorphic to a smooth Fano threefold. We prove the main result in two steps. In the first step, we deduce from results of Goertsches, Konstantis, and Zoller that if the complexity of the action is zero or one then the equivariant and the ordinary cohomology with integer coefficients are determined by the GKM graph. This result, in combination with a classification result by Jupp, Wall and Zubr for certain six-manifolds, implies that the diffeomorphism type of a compact symplectic six-manifold with a Hamiltonian GKM action is determined by the associated GKM graph. In the second step, based on results by Godinho and Sabatini, we compute the complete list of the GKM graphs of positive Hamiltonian GKM spaces of dimension six. We deduce that any such GKM graph is isomorphic to a GKM graph of a smooth Fano threefold.

math.SG↗

arXiv2023

Compact monotone tall complexity one $T$-spaces

In this paper we study compact monotone tall complexity one $T$-spaces. We use the classification of Karshon and Tolman, and the monotone condition, to prove that any two such spaces are isomorphic if and only if they have equal Duistermaat-Heckman measures. Moreover, we show that the moment polytope is Delzant and reflexive, and provide a complete description of the possible Duistermaat-Heckman measures. Whence we obtain a finiteness result that is analogous to that for compact monotone symplectic toric manifolds. Furthermore, we show that any such $T$-action can be extended to a toric $(T \times S^1)$-action. Motivated by a conjecture of Fine and Panov, we prove that any compact monotone tall complexity one $T$-space is equivariantly symplectomorphic to a Fano manifold endowed with a suitable symplectic form and a complexity one $T$-action.

math.SG↗

arXiv2020

Toric one-skeletons for complexity-one spaces

A complexity-one space is a compact symplectic manifold $(M, ω)$ endowed with an effective Hamiltonian action of a torus $T$ of dimension $\frac{1}{2}\dim(M)-1$. In this note we prove that for a certain class of complexity-one spaces the Poincaré dual of the Chern class $c_{n-1}$ can be represented by a collection of $\frac{n}{2}χ(M)$ symplectic embedded $2$-spheres, where $χ(M)$ is the Euler characteristic of $M$ and $\dim(M)=2n$. We call such a collection a toric one-skeleton. The classification of complexity-one spaces is an important subject in symplectic geometry. A nice subcategory of those spaces are the ones which are monotone. The existence of a toric one-skeleton is a useful tool to understand six-dimensional monotone complexity-one spaces. In particular, we will show that the existence of a toric one-skeleton for such a space implies that the second Betti number of $M$ is at most seven. This is a simple application of results by Sabatini-Sepe and Lindsay-Panov.

math.AT↗

arXiv2019

Hamiltonian S^1-spaces with large equivariant pseudo-index

Let $(M,ω)$ be a compact symplectic manifold of dimension $2n$ endowed with a Hamiltonian circle action with only isolated fixed points. Whenever $M$ admits a toric $1$-skeleton $\mathcal{S}$, which is a special collection of embedded $2$-spheres in $M$, we define the notion of equivariant pseudo-index of $\mathcal{S}$: this is the minimum of the evaluation of the first Chern class $c_1$ on the spheres of $\mathcal{S}$. This can be seen as the analog in this category of the notion of pseudo-index for complex Fano varieties. In this paper we provide upper bounds for the equivariant pseudo-index. In particular, when the even Betti numbers of $M$ are unimodal, we prove that it is at most $n+1$ . Moreover, when it is exactly $n+1$, $M$ must be homotopically equivalent to $\C P^n$.

math.AT↗