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Susan Tolman

Publications and source records attributed to Susan Tolman.

At least 19 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.

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arXiv2025

Connectedness of fibers beyond semitoric systems II: ephemeral singular points

In an earlier paper, we proved the connectedness of the fibers of every $2n$-dimensional integrable system satisfying both: the action extends the action of an $(n-1)$-dimensional torus which has a proper moment map, and every tall singular point is non-degenerate and no such point has a hyperbolic block and connected $T$-stabilizer. Unfortunately, these criteria are fairly restrictive. Our main goal in this paper is to find a larger class of integrable systems that has connected fibers by weakening the non-degeneracy assumption above. To achieve this, we introduce ``ephemeral" degenerate singular points, examples of which have appeared in the literature in the context of both $p \! : \! -q$ resonances and special Lagrangian fibrations. Finally, we construct a family of examples that shows that our main theorem meaningfully extends previous results.

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arXiv2024

Equivariant cohomological rigidity for four-dimensional Hamiltonian $\mathbf{S^1}$-manifolds

For manifolds equipped with group actions, we have the following natural question: To what extent does the equivariant cohomology determine the equivariant diffeotype? We resolve this question for Hamiltonian circle actions on compact, connected symplectic four-manifolds. They are equivariantly diffeomorphic if and only if their equivariant cohomology rings are isomorphic as algebras over the equivariant cohomology of a point. In fact, we prove a stronger claim: each isomorphism between their equivariant cohomology rings is induced by an equivariant diffeomorphism.

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arXiv2024

Connectedness of fibers beyond semitoric systems I: the non-degenerate case

In this paper we study the connectedness of the fibers of integrable systems that extend complexity one $T$-spaces with proper moment maps, assuming that every tall singular point is non-degenerate. Our main result states that if there are no tall singular points with a hyperbolic block and connected $T$-stabilizer, then each fiber is connected. Moreover, we prove that the above condition is necessary if either some reduced space is simply connected or the moment map for the integrable system is generic in a natural sense.

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arXiv2020

Non-Hamiltonian actions with fewer isolated fixed points

In an earlier paper, the second author resolved a question of McDuff by constructing a non-Hamiltonian symplectic circle action on a closed, connected six-dimensional symplectic manifold with exactly 32 fixed points. In this paper, we improve on this example by reducing the number of fixed points. More concretely, we construct a non-Hamiltonian symplectic circle action on a closed, connected six-dimensional symplectic manifold with exactly $2k$ fixed points for any $k \geq 5$.

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arXiv2020

Topology of complexity one quotients

We describe of the topology of the geometric quotients of 2n dimensional compact connected symplectic manifolds with n-1 dimensional torus actions. When the isotropy weights at each fixed point are in general position, the quotient is homeomorphic to a sphere.

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arXiv2020

Symplectic cohomological rigidity via toric degnerations

In this paper we study whether symplectic toric manifolds are symplectically cohomologically rigid. Here we say that symplectic cohomological rigidity holds for some family of symplectic manifolds if the members of that family can be distinguished by their integral cohomology rings and the cohomology classes of their symplectic forms. We show how toric degenerations can be used to produce the symplectomorphisms necessary to answer this question. As a consequence we prove that symplectic cohomological rigidity holds for the family of symplectic Bott manifolds with rational symplectic form whose rational cohomology ring is isomorphic to $\mathrm{H}^*((\mathbb{CP}^1)^n;\mathbb{Q})$ for some $n$. In particular, we classify such manifolds up to symplectomorphism. Moreover, we prove that any symplectic toric manifold with rational symplectic form whose integral cohomology ring is isomorphic to $\mathrm{H}^*((\mathbb{CP}^1)^n;\mathbb{Z})$ is symplectomorphic to $(\mathbb{CP}^1)^n$ with a product symplectic structure.

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arXiv2017

Tame Circle Actions

In this paper, we consider Sjamaar's holomorphic slice theorem, the birational equivalence theorem of Guillemin and Sternberg, and a number of important standard constructions that work for Hamiltonian circle actions in both the symplectic category and the Kähler category: reduction, cutting, and blow-up. In each case, we show that the theory extends to Hamiltonian circle actions on complex manifolds with tamed symplectic forms. (At least, the theory extends if the fixed points are isolated.) Our main motivation for this paper is that the first author needs the machinery that we develop here to construct a non-Hamiltonian symplectic circle action on a closed, connected six-dimensional symplectic manifold with exactly 32 fixed points; this answers an open question in symplectic geometry. However, we also believe that the setting we work in is intrinsically interesting, and elucidates the key role played by the following fact: the moment image of $e^t \cdot x$ increases as $t \in \mathbb{R}$ increases.

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arXiv2016

Hamiltonian circle actions on eight dimensional manifolds with minimal fixed sets

Consider a Hamiltonian circle action on a closed $8$-dimensional symplectic manifold $M$ with exactly five fixed points, which is the smallest possible fixed set. In their paper, L. Godinho and S. Sabatini show that if $M$ satisfies an extra "positivity condition" then the isotropy weights at the fixed points of $M$ agree with those of some linear action on $\mathbb{CP}^4$. Therefore, the (equivariant) cohomology rings and the (equivariant) Chern classes of $M$ and $\mathbb{CP}^4$ agree; in particular, $H^*(M;\mathbb{Z}) \simeq \mathbb{Z}[y]/y^5$ and $c(TM) = (1+y)^5$. In this paper, we prove that this positivity condition always holds for these manifolds. This completes the proof of the "symplectic Petrie conjecture" for Hamiltonian circle actions on on 8-dimensional closed symplectic manifolds with minimal fixed sets.

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arXiv2012

Hamiltonian circle actions with minimal fixed sets

Consider an effective Hamiltonian circle action on a compact symplectic $2n$-dimensional manifold $(M, ω)$. Assume that the fixed set $M^{S^1}$ is {\em minimal}, in two senses: it has exactly two components, $X$ and $Y$, and $\dim(X) + \dim(Y) = \dim(M) - 2$. We prove that the integral cohomology ring and Chern classes of $M$ are isomorphic to either those of $\CP^n$ or (if $n \neq 1$ is odd) to those of $\Gt_2(\R^{n+2})$, the Grassmannian of oriented two-planes in $\R^{n+2}$. In particular, $H^i(M;\Z) = H^i(\CP^n;\Z)$ for all $i$, and the Chern classes of $M$ are determined by the integral cohomology {\em ring}. We also prove that the fixed set data of $M$ agrees exactly with the fixed set data for one of the standard circle actions on one of these two manifolds. In particular, we show that there are no points with stabilizer $\Z_k$ for any $k > 2$. The same conclusions hold when $M^{S^1}$ has exactly two components and the even Betti numbers of $M$ are minimal, that is, $b_{2i}(M) = 1$ for all $i \in \{0,...,1/2\dim(M)\}$. This provides additional evidence that very few symplectic manifolds with minimal even Betti numbers admit Hamiltonian actions.

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arXiv2012

New Techniques for obtaining Schubert-type formulas for Hamiltonian manifolds

In [GT], Goldin and the second author extend some ideas from Schubert calculus to the more general setting of Hamiltonian torus actions on compact symplectic manifolds with isolated fixed points. (See also [Kn99] and [Kn08].) The main goal of this paper is to build on this work by finding more effective formulas. More explicitly, given a generic component of the moment map, they define a canonical class $α_p$ in the equivariant cohomology of the manifold $M$ for each fixed point $p \in M$. When they exist, canonical classes form a natural basis of the equivariant cohomology of $M$. In particular, when $M$ is a flag variety, these classes are the equivariant Schubert classes. It is a long standing problem in combinatorics to find positive integral formulas for the equivariant structure constants associated to this basis. Since computing the restriction of the canonical classes to the fixed points determines these structure constants, it is important to find effective formulas for these restrictions. In this paper, we introduce new techniques for calculating the restrictions of a canonical class $α_p$ to a fixed point $q$. Our formulas are nearly always simpler, in the sense that they count the contributions over fewer paths. Moreover, our formula is manifestly positive and integral in certain important special cases.

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arXiv2012

Classification of Hamiltonian torus actions with two dimensional quotients

We construct all possible Hamiltonian torus actions for which all the non-empty reduced spaces are two dimensional (and not single points) and the manifold is connected and compact, or, more generally, the moment map is proper as a map to a convex set. This construction completes the classification of tall complexity one spaces.

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arXiv2011

Polytopes with mass linear functions II: the 4-dimensional case

This paper continues the analysis begun in {\it Polytopes with mass linear functions, Part I} of the structure of smooth moment polytopes $Δ\subset \ft^*$ that support a mass linear function $H \in \ft$. As explained there, besides its purely combinatorial interest, this question is relevant to the study of the homomorphism $π_1(T^n)\to π_1(\Symp(M_Δ, ω_Δ))$ from the fundamental group of the torus $T^n$ to that of the group of symplectomorphisms of the $2n$-dimensional symplectic toric manifold $(M_Δ, ω_Δ)$ associated to $Δ$. In Part I, we made a general investigation of this question and classified all mass linear pairs $(Δ, H)$ in dimensions up to three. The main result of the current paper is a classification of all 4-dimensional examples. Along the way, we investigate the properties of general constructions such as fibrations, blowups and expansions (or wedges), describing their effect both on moment polytopes and on mass linear functions. We end by discussing the relation of mass linearity to Shelukhin's notion of full mass linearity. The two concepts agree in dimensions up to and including 4. However full mass linearity may be the more natural concept when considering the question of which blow ups preserve mass linearity.

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arXiv2010

Toric integrable geodesic flows in odd dimensions

Let $Q$ be a compact, connected $n$-dimensional Riemannian manifold, and assume that the geodesic flow is toric integrable. If $n \neq 3$ is odd, or if $\pi_1(Q)$ is infinite, we show that the cosphere bundle of $Q$ is equivariantly contactomorphic to the cosphere bundle of the torus $\T^n$. As a consequence, $Q$ is homeomorphic to $\T^n$.

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arXiv2010

Fixed points of symplectic periodic flows

The study of fixed points is a classical subject in geometry and dynamics. If the circle acts in a Hamiltonian fashion on a compact symplectic manifold M, then it is classically known that there are at least 1 + dim(M)/2 fixed points; this follows from Morse theory for the momentum map of the action. In this paper we use Atiyah-Bott-Berline-Vergne (ABBV) localization in equivariant cohomology to prove that this conclusion also holds for symplectic circle actions with non-empty fixed sets, as long as the Chern class map is somewhere injective -- the Chern class map assigns to a fixed point the sum of the action weights at the point. We complement this result with less sharp lower bounds on the number of fixed points, under no assumptions; from a dynamical systems viewpoint, our results imply that there is no symplectic periodic flow with exactly one or two equilibrium points on a compact manifold of dimension at least eight.

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arXiv2009

Polytopes with mass linear functions, part I

We analyze mass linear functions $H$ on simple polytopes $\De$, where a mass linear function is an affine function on $\De$ whose value on the center of mass depends linearly on the positions of the supporting hyperplanes. We show that certain types of symmetries of $\De$ give rise to nonconstant mass linear functions on $\De$. These are called inessential; the others are essential. We also show that most polytopes do not admit any nonconstant mass linear functions. Our main result shows that there is only one family of smooth polytopes of dimension $\leq 3$ which admit essential mass linear functions. These results have geometric implications. Fix a symplectic toric manifold $(M,\om,T,Φ)$ with moment polytope $\De = Φ(M)$; let $\Symp(M,\om)$ be its group of symplectomorphisms. Any linear function $H$ on $\De$ generates a Hamiltonian $\R$ action on $M$ whose closure is a subtorus $T_H$ of $T$. We show that if the map $π_1(T_H)\to π_1(\Symp(M,\om))$ has finite image, then $H$ is mass linear. Therefore, in most cases the induced map $π_1(T) \to π_1(\Symp(M,\om))$ is an injection. We also show that this map does not have finite image unless $M$ is a product of projective spaces. Moreover, the inessential $H$ correspond to elements in the kernel of the map $π_1(T)\to \Isom(M)$, where the Kahler isometry group $\Isom(M)\subset \Symp(M,\om)$ consists of elements that also preserve the natural compatible complex structure on $M$. Therefore if $\De$ supports no nonconstant essential mass linear $H$, the map $π_1(\Isom(M))\to pi_1(\Symp(M,\om)$ is injective.

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arXiv2009

On a symplectic generalization of Petrie's conjecture

Motivated by the Petrie conjecture, we consider the following questions: Let a circle act in a Hamiltonian fashion on a compact symplectic manifold $(M,ω)$ which satisfies $H^{2i}(M;\R) = H^{2i}(\CP^n,\R)$ for all $i$. Is $H^j(M;\Z) = H^j(\CP^n;\Z)$ for all $j$? Is the total Chern class of $M$ determined by the cohomology ring $H^*(M;\Z)$? We answer these questions in the six dimensional case by showing that $H^j(M;\Z)$ is equal to $H^j(\CP^3;\Z)$ for all $j$, by proving that only four cohomology rings can arise, and by computing the total Chern class in each case. We also prove that there are no exotic actions. More precisely, if $H^*(M;\Z)$ is isomorphic to $H^*(\CP^3;\Z)$ or $H^*(\Tilde{G}_2(\R^5);\Z)$, then the representations at the fixed components are compatible with one of the standard actions; in the remaining two case, the representation is strictly determined by the cohomology ring. Finally, our results suggest a natural question: do the remaining two cohomology rings actually arise? This question is closely related to some interesting problems in symplectic topology, such as embeddings of ellipsoids.

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