arXiv subjects

Dusa McDuff

Publications and source records attributed to Dusa McDuff.

At least 37 records · Page 2Linked to original sources

arXiv2015

Smooth Kuranishi atlases with isotropy

Kuranishi structures were introduced in the 1990s by Fukaya and Ono for the purpose of assigning a virtual cycle to moduli spaces of pseudoholomorphic curves that cannot be regularized by geometric methods. Their core idea was to build such a cycle by patching local finite dimensional reductions, given by smooth sections that are equivariant under a finite isotropy group. Building on our notions of topological Kuranishi atlases and perturbation constructions in the case of trivial isotropy, we develop a theory of Kuranishi atlases and cobordisms that transparently resolves the challenges posed by nontrivial isotropy. We assign to a cobordism class of weak Kuranishi atlases both a virtual moduli cycle (VMC - a cobordism class of weighted branched manifolds) and a virtual fundamental class (VFC - a Cech homology class).

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arXiv2015

Symplectic embeddings of 4-dimensional ellipsoids: erratum

This note describes a correct way to perform the inflation procedures claimed in the papers on embedding ellipsoids, Journ. Top. 2 (2009), 1-22 and 589-623. The idea is to inflate along a collection of transversally and positively intersecting symplectic surfaces (as in Li--Usher (JSG 2006)) rather than along a single embedded surface.

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arXiv2013

Nongeneric J-holomorphic curves and singular inflation

This paper investigates the geometry of a symplectic 4-manifold $(M,\om)$ relative to a J-holomorphic normal crossing divisor S. Extending work by Biran (in Invent. Math. 1999), we give conditions under which a homology class $A\in H_2(M;\Z)$ with nontrivial Gromov invariant has an embedded J-holomorphic representative for some S-compatible J. This holds for example if the class $A$ can be represented by an embedded sphere, or if the components of S are spheres with self-intersection -2. We also show that inflation relative to S is always possible, a result that allows one to calculate the relative symplectic cone. It also has important applications to various embedding problems, for example of ellipsoids or Lagrangian submanifolds.

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arXiv2013

Displacing Lagrangian toric fibers by extended probes

In this paper we introduce a new way of displacing Lagrangian fibers in toric symplectic manifolds, a generalization of McDuff's original method of probes. Extended probes are formed by deflecting one probe by another auxiliary probe. Using them, we are able to displace all fibers in Hirzebruch surfaces except those already known to be nondisplaceable, and can also displace an open dense set of fibers in the weighted projective space P(1,3,5) after resolving the singularities. We also investigate the displaceability question in sectors and their resolutions. There are still many cases in which there is an open set of fibers whose displaceability status is unknown.

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arXiv2013

The Gromov width of 4-dimensional tori

We show that every 4-dimensional torus with a linear symplectic form can be fully filled by one symplectic ball. If such a torus is not symplectomorphic to a product of 2-dimensional tori with equal sized factors, then it can also be fully filled by any finite collection of balls provided only that their total volume is less than that of the 4-torus with its given linear symplectic form.

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arXiv2013

Nongeneric J-holomorphic curves in symplectic 4-manifolds

This note discusses the structure of J-holomorphic curves in symplectic 4-manifolds (M,\om) when J\in \Jj(\Ss), the set of \om-tame J for which a fixed chain \Ss of transversally intersecting embedded spheres of self-intersection \le -2 is J-holomorphic. Extending work by Biran (in Invent. Math. (1999)), it shows that when (M,\om) is the blow up of a rational or ruled symplectic 4-manifold, any homology class A\in H_2(M;\Z), with nonzero Gromov invariant and nonnegative intersection both with the spheres in \Ss and with the exceptional classes other than A, has an embedded J-holomorphic representative for some J\in \Jj(\Ss$. This result is a key step in some of the arguments in McDuff (Journ. Topology (2009)) on embedding ellipsoids, and also has applications to symplectic 4-orbifolds.

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arXiv2011

Displacing Lagrangian toric fibers via probes

This note studies the geometric structure of monotone moment polytopes (the duals of smooth Fano polytopes) using probes. The latter are line segments that enter the polytope at an interior point of a facet and whose direction is integrally transverse to this facet. A point inside the polytope is displaceable by a probe if it lies less than half way along it. Using a construction due to Fukaya-Oh-Ohta-Ono, we show that every rational polytope has a central point that is not displaceable by probes. In the monotone (or more generally, the reflexive) case, this central point is its unique interior integral point. In the monotone case, every other point is displaceable by probes if and only if the polytope satisfies the star Ewald condition. (This is a strong version of the Ewald conjecture concerning the integral symmetric points in the polytope.) Further, in dimensions up to and including three every monotone polytope is star Ewald. These results are closely related to the Fukaya-Oh-Ohta-Ono calculations of the Floer homology of the Lagrangian fibers of a toric symplectic manifold, and have applications to questions introduced by Entov-Polterovich about the displaceability of these fibers.

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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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arXiv2011

The Hofer conjecture on embedding symplectic ellipsoids

In this note we show that one open four dimensional ellipsoid embeds symplectically into another if and only the ECH capacities of the first are no larger than those of the second. This proves a conjecture due to Hofer. The argument uses the equivalence of the ellipsoidal embedding problem with a ball embedding problem that was recently established by McDuff. Its method is inspired by Hutchings' recent results on embedded contact homology (ECH) capacities but does not use them.

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arXiv2010

The topology of toric symplectic manifolds

This is a collection of results on the topology of toric symplectic manifolds. Using an idea of Borisov, we show that a closed symplectic manifold supports at most a finite number of toric structures. Further, the product of two projective spaces of complex dimension at least two (and with a standard product symplectic form) has a unique toric structure. We then discuss various constructions, using wedging to build a monotone toric symplectic manifold whose center is not the unique point displaceable by probes, and bundles and blow ups to form manifolds with more than one toric structure. The bundle construction uses the McDuff--Tolman concept of mass linear function. Using Timorin's description of the cohomology ring via the volume function we develop a cohomological criterion for a function to be mass linear, and explain its relation to Shelukhin's higher codimension barycenters.

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arXiv2010

Monodromy in Hamiltonian Floer theory

Schwarz showed that when a closed symplectic manifold (M,\om) is symplectically aspherical (i.e. the symplectic form and the first Chern class vanish on π_2(M)) then the spectral invariants, which are initially defined on the universal cover of the Hamiltonian group, descend to the Hamiltonian group Ham (M,\om). In this note we describe less stringent conditions on the Chern class and quantum homology of M under which the (asymptotic) spectral invariants descend to Ham (M,\om). For example, they descend if the quantum multiplication of M is undeformed and H_2(M) has rank >1, or if the minimal Chern number is at least n+1 (where \dim M=2n) and the even cohomology of M is generated by divisors. The proofs are based on certain calculations of genus zero Gromov--Witten invariants. As an application, we show that the Hamiltonian group of the one point blow up of T^4 admits a Calabi quasimorphism. Moreover, whenever the (asymptotic) spectral invariants descend it is easy to see that Ham (M,\om) has infinite diameter in the Hofer norm. Hence our results establish the infinite diameter of Ham in many new cases. We also show that the area pseudonorm -- a geometric version of the Hofer norm -- is nontrivial on the (compactly supported) Hamiltonian group for all noncompact manifolds as well as for a large class of closed manifolds.

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arXiv2010

The embedding capacity of 4-dimensional symplectic ellipsoids

This paper calculates the function $c(a)$ whose value at $a$ is the infimum of the size of a ball that contains a symplectic image of the ellipsoid $E(1,a)$. (Here $a \ge 1$ is the ratio of the area of the large axis to that of the smaller axis.) The structure of the graph of $c(a)$ is surprisingly rich. The volume constraint implies that $c(a)$ is always greater than or equal to the square root of $a$, and it is not hard to see that this is equality for large $a$. However, for $a$ less than the fourth power $τ^4$ of the golden ratio, $c(a)$ is piecewise linear, with graph that alternately lies on a line through the origin and is horizontal. We prove this by showing that there are exceptional curves in blow ups of the complex projective plane whose homology classes are given by the continued fraction expansions of ratios of Fibonacci numbers. On the interval $[τ^4,7]$ we find $c(a)=(a+1)/3$. For $a \ge 7$, the function $c(a)$ coincides with the square root except on a finite number of intervals where it is again piecewise linear. The embedding constraints coming from embedded contact homology give rise to another capacity function $c_{ECH}$ which may be computed by counting lattice points in appropriate right angled triangles. According to Hutchings and Taubes, the functorial properties of embedded contact homology imply that $c_{ECH}(a) \le c(a)$ for all $a$. We show here that $c_{ECH}(a) \ge c(a)$ for all $a$.

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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

Symplectic embeddings and continued fractions: a survey

As has been known since the time of Gromov's Nonsqueezing Theorem, symplectic embedding questions lie at the heart of symplectic geometry. After surveying some of the most important ways of measuring the size of a symplectic set, these notes discuss some recent developments concerning the question of when a 4-dimensional ellipsoid can be symplectically embedded in a ball. This problem turns out to have unexpected relations to the properties of continued fractions and of exceptional curves in blow ups of the complex projective plane. It is also related to questions of lattice packing of planar triangles.

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arXiv2009

Some 6-dimensional Hamiltonian S^1 manifolds

In an earlier paper we explained how to convert the problem of symplectically embedding one 4-dimensional ellipsoid into another into the problem of embedding a certain set of disjoint balls into \CP^2 by using a new way to desingularize orbifold blow ups Z of the weighted projective space \CP^2_{1,m,n}. We now use a related method to construct symplectomorphisms of these spaces Z. This allows us to construct some well known Fano 3-folds (including the Mukai--Umemura 3-fold) in purely symplectic terms, using a classification by Tolman of a particular class of Hamiltonian S^1-manifolds. We also show that these manifolds are uniquely determined by their fixed point data up to equivariant symplectomorphism. As part of this argument we show that the symplectomorphism group of a certain weighted blow up of a weighted projective plane is connected.

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arXiv2009

Hamiltonian S^1 manifolds are uniruled

The main result of this note is that every closed Hamiltonian S^1 manifold is uniruled, i.e. it has a nonzero Gromov--Witten invariant one of whose constraints is a point. The proof uses the Seidel representation of π_1 of the Hamiltonian group in the small quantum homology of M as well as the blow up technique recently introduced by Hu, Li and Ruan. It applies more generally to manifolds that have a loop of Hamiltonian symplectomorphisms with a nondegenerate fixed maximum. Some consequences for Hofer geometry are explored. An appendix discusses the structure of the quantum homology ring of uniruled manifolds.

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arXiv2009

Loops in the Hamiltonian group: a survey

This note describes some recent results about the homotopy properties of Hamiltonian loops in various manifolds, including toric manifolds and one point blow ups. We describe conditions under which a circle action does not contract in the Hamiltonian group, and construct an example of a loop $\ga$ of diffeomorphisms of a symplectic manifold M with the property that none of the loops smoothly isotopic to $\ga$ preserve any symplectic form on M. We also discuss some new conditions under which the Hamiltonian group has infinite Hofer diameter. Some of the methods used are classical (Weinstein's action homomorphism and volume calculations), while others use quantum methods (the Seidel representation and spectral invariants).

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arXiv2008

Symplectic embeddings of 4-dimensional ellipsoids

We show how to reduce the problem of symplectically embedding one 4-dimensional rational ellipsoid into another to a problem of embedding disjoint unions of balls into appropriate blow ups of \C P^2. For example, the problem of embedding the ellipsoid E(1,k) into a ball B is equivalent to that of embedding k disjoint equal balls into \C P^2, and so can be solved by the work of Gromov, McDuff--Polterovich and Biran. (Here k is the ratio of the area of the major axis to that of the minor axis.) As a consequence we show that the ball may be fully filled by the ellipsoid E(1,k) for k=1,4 and all k\ge 9, thus answering a question raised by Hofer.

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