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Liat Kessler

Publications and source records attributed to Liat Kessler.

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.

math.SG↗
arXiv2025

On the equivariant cohomological rigidity of semi-free Hamiltonian circle actions

We consider semi-free Hamiltonian $S^1$-manifolds of dimension six and establish when the equivariant cohomology and data on the fixed point set determine the isomorphism type. Gonzales listed conditions under which the isomorphism type of such spaces is determined by fixed point data. We pointed out in an earlier paper that this result as stated is erroneous, and proved a corrected version. However, that version relied on a certain distribution of fixed points that is not at all necessary. In this paper, we replace the latter assumption with a global assumption on equivariant cohomology that is necessary for an isomorphism. We also extend our result to the equivariant (non-symplectic) topological category. The variation in the earlier paper was tailored to suit the requirements of Cho's application of Gonzales' statement to classify semi-free monotone, Hamiltonian $S^1$-manifolds of dimension six. In the current paper, we aim to give the definitive statement relating fixed point data and equivariant cohomology to the isomorphism type of a semi-free Hamiltonian $S^1$-manifold.

math.SG↗
arXiv2025

On isomorphisms of semi-free Hamiltonian $S^1$-manifolds and fixed point data

Following Gonzales, we answer the question of whether the isomorphism type of a semi-free Hamiltonian $S^1$-manifold of dimension six is determined by certain data on the critical levels. We first give counter examples showing that Gonzales' assumptions are not sufficient for a positive answer. Then we prove that it is enough to further assume that the reduced spaces of dimension four are symplectic rational surfaces and the interior fixed surfaces are restricted to at most one level. The additional assumptions allow us to use results proven by $J$-holomorphic methods. Gonzales' answer was applied by Cho in proving that if the underlying symplectic manifold is positive monotone then the space is isomorphic to a Fano manifold with a holomorphic $S^1$-action. We show that our variation is enough for Cho's application.

math.SG↗
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.

math.SG↗
arXiv2023

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

Let $(M,\omega)$ 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.

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arXiv2019

Equivariant cohomology of a complexity-one four-manifold is determined by combinatorial data

For Hamiltonian circle actions on compact, connected, four-dimensional manifolds, we give a generators and relations description for the even part of the equivariant cohomology, as an algebra over the equivariant cohomology of a point. This description depends on combinatorial data encoded in the decorated graph of the manifold. We then give an explicit combinatorial description of all weak algebra isomorphisms. We use this description to prove that the even parts of the equivariant cohomology algebras are weakly isomorphic and the odd groups have the same ranks if and only if the labeled graphs obtained from the decorated graphs by forgetting the height and area labels are isomorphic. As a consequence, we give an example of an isomorphism of equivariant cohomology algebras that cannot be induced by an equivariant diffeomorphism of manifolds preserving a compatible almost complex structure. We also provide a soft proof that there are finitely many maximal Hamiltonian circle actions on a fixed compact, connected, four-dimensional symplectic manifold.

math.SG↗
arXiv2019

The equivariant cohomology of complexity one spaces

Complexity one spaces are an important class of examples in symplectic geometry. Karshon and Tolman classify them in terms of combinatorial and topological data. In this paper, we compute the equivariant cohomology for any complexity one space $T^{n-1}$ acting on $M^{2n}$. The key step is to compute the equivariant cohomology for any Hamiltonian $S^1$ action on $M^4$.

math.SG↗
arXiv2017

Circle actions on symplectic four-manifolds

We complete the classification of Hamiltonian torus and circle actions on symplectic four-dimensional manifolds. Following work of Delzant and Karshon, Hamiltonian circle and 2-torus actions on any fixed simply connected symplectic four-manifold were characterized by Karshon, Kessler and Pinsonnault. What remains is to study the case of Hamiltonian actions on blowups of S^2-bundles over a Riemann surface of positive genus. These do not admit 2-torus actions. In this paper, we characterize Hamiltonian circle actions on them. We then derive combinatorial results on the existence and counting of these actions. As a by-product, we provide an algorithm that determines the g-reduced form of a blowup form. Our work is a combination of "soft" equivariant and combinatorial techniques, using the momentum map and related data, with "hard" holomorphic techniques, including Gromov-Witten invariants.

math.SG↗
arXiv2014

Non-Noetherianity of Denjoy-Carleman rings of germs

It is shown that Denjoy-Carleman quasi analytic rings of germs of functions in two or more variables either complex or real valued that are stable under derivation and strictly larger than the ring of real-analytic germs are not Noetherian rings. The failure of Weierstrass division on these Denjoy-Carleman classes yields a contradiction to Noetherianity via a stronger version of Artin Approximation due to Popescu as well as results on projective modules. This settles a 35-year old open problem in real algebraic geometry.

math.AG↗
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.

math.SG↗
arXiv2014

Distinguishing symplectic blowups of the complex projective plane

A symplectic manifold that is obtained from the complex projective plane by k blowups is encoded by k+1 parameters: the size of the initial complex projective plane, and the sizes of the blowups. We determine which values of these parameters yield symplectomorphic manifolds.

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arXiv2011

Symplectic forms on the space of embedded symplectic surfaces and their reductions

Let (M,ω) be a symplectic manifold, and (Σ,σ) a closed connected symplectic 2-manifold. We construct a weakly symplectic form {ω^{D}}_{(Σ, σ)} on the space of immersions Σ\to M that is a special case of Donaldson's form. We show that the restriction of {ω^{D}}_{(Σ,σ)} to any orbit of the group of Hamiltonian symplectomorphisms through a symplectic embedding (Σ,σ) \to (M,ω) descends to a weakly symplectic form ω^D_{\red} on the quotient by Sympl(Σ,σ), and that the obtained symplectic space is a symplectic quotient of the subspace of symplectic embeddings S_{e}(Σ,σ) with respect to the Sympl(Σ,σ)-action. We also compare {ω^{D}}_{(Σ,σ)} and its reduction ω^D_{\red} to another 2-form on the space of immersed symplectic Σ-surfaces in M. We conclude by a result on the restriction of {ω^{D}}_{(Σ,σ)} to moduli spaces of J-holomorphic curves.

math.SG↗
arXiv2010

Symplectic geometry on moduli spaces of J-holomorphic curves

Let (M,ω) be a symplectic manifold, and Sigma a compact Riemann surface. We define a 2-form on the space of immersed symplectic surfaces in M, and show that the form is closed and non-degenerate, up to reparametrizations. Then we give conditions on a compatible almost complex structure J on (M,ω) that ensure that the restriction of the form to the moduli space of simple immersed J-holomorphic Sigma-curves in a homology class A in H_2(M,\Z) is a symplectic form, and show applications and examples. In particular, we deduce sufficient conditions for the existence of J-holomorphic Sigma-curves in a given homology class for a generic J.

math.SG↗
arXiv2009

Holomorphic shadows in the eyes of model theory

We define a subset of an almost complex manifold (M,J) to be a holomorphic shadow if it is the image of a J-holomorphic map from a compact complex manifold. Notice that a J-holomorphic curve is a holomorphic shadow, and so is a complex subvariety of a compact complex manifold. We show that under some conditions on an almost complex structure J on a manifold M, the holomorphic shadows in the Cartesian products of (M,J) form a Zariski-type structure. Checking this leads to non-trivial geometric questions and results. We then apply the work of Hrushovski and Zilber on Zariski-type structures. We also restate results of Gromov and McDuff on J-holomorphic curves in symplectic geometry in the language of shadows structures.

math.DG↗
arXiv2008

Torus actions on small blow ups of CP^2

A manifold obtained by k simultaneous symplectic blow-ups of CP^2 of equal sizes epsilon (where the size of CP^1 in CP^2 is one) admits an effective two-dimensional torus action if k <= 3. We show that it does not admit such an action if k >=4 and epsilon <= 1/(3k 2^{2k}). For the proof, we correspond between the geometry of a symplectic toric four-manifold and the combinatorics of its moment map image. We also use techniques from the theory of J-holomorphic curves.

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.

math.SG↗