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Lagrangian Floer theory on compact toric manifolds I
, 2009
"... The present authors introduced the notion of weakly unobstructed Lagrangian submanifolds and constructed their potential function PO purely in terms of Amodel data in [FOOO3]. In this paper, we carry out explicit calculations involving PO on toric manifolds and study the relationship between this ..."
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Cited by 21 (3 self)
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The present authors introduced the notion of weakly unobstructed Lagrangian submanifolds and constructed their potential function PO purely in terms of Amodel data in [FOOO3]. In this paper, we carry out explicit calculations involving PO on toric manifolds and study the relationship between this class of Lagrangian submanifolds with the earlier work of Givental [Gi1] which advocates that quantum cohomology ring is isomorphic to the Jacobian ring of a certain function, called the LandauGinzburg superpotential. Combining this study with the results from [FOOO3], we also apply the study to various examples to illustrate its implications to symplectic topology of Lagrangian fibers of toric manifolds. In particular we relate it to Hamiltonian displacement property of Lagrangian fibers and to EntovPolterovich’s
On the extrinsic topology of Lagrangian submanifolds
, 2005
"... Abstract. We investigate the extrinsic topology of Lagrangian submanifolds and of their submanifolds in closed symplectic manifolds using Floer homological methods. The first result asserts that the homology class of a displaceable monotone Lagrangian submanifold vanishes in the homology of the ambi ..."
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Cited by 12 (1 self)
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Abstract. We investigate the extrinsic topology of Lagrangian submanifolds and of their submanifolds in closed symplectic manifolds using Floer homological methods. The first result asserts that the homology class of a displaceable monotone Lagrangian submanifold vanishes in the homology of the ambient symplectic manifold. Combining this with spectral invariants we provide a new mechanism for proving Lagrangian intersection results e.g. entailing that any two simply connected Lagrangian submanifold in CP n ×CP n must intersect. Contents
Calabi quasimorphisms for some nonmonotone symplectic manifolds
 Algebr. Geom. Topol
"... In this work we construct Calabi quasimorphisms on the universal cover ˜Ham(M) of the group of Hamiltonian diffeomorphisms for some nonmonotone symplectic manifolds. This complements a result by Entov and Polterovich which applies in the monotone case. Moreover, in contrast with their work, we sho ..."
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Cited by 10 (1 self)
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In this work we construct Calabi quasimorphisms on the universal cover ˜Ham(M) of the group of Hamiltonian diffeomorphisms for some nonmonotone symplectic manifolds. This complements a result by Entov and Polterovich which applies in the monotone case. Moreover, in contrast with their work, we show that these quasimorphisms descend to nontrivial homomorphisms on the fundamental group of Ham(M). 1
Spectral numbers in Floer theories
"... Abstract. The chain complexes underlying Floer homology theories typically carry a realvalued filtration, allowing one to associate to each Floer homology class a spectral number defined as the infimum of the filtration levels of chains representing that class. These spectral numbers have been stud ..."
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Cited by 7 (2 self)
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Abstract. The chain complexes underlying Floer homology theories typically carry a realvalued filtration, allowing one to associate to each Floer homology class a spectral number defined as the infimum of the filtration levels of chains representing that class. These spectral numbers have been studied extensively in the case of Hamiltonian Floer homology by Oh, Schwarz, and others. We prove that the spectral number associated to any nonzero Floer homology class is always finite, and that the infimum in the definition of the spectral number is always attained. In the Hamiltonian case, this implies that what is known as the “nondegenerate spectrality ” axiom holds on all closed symplectic manifolds. Our proofs are entirely algebraic and rather elementary, and apply to any Floertype theory (including Novikov homology) satisfying certain standard formal properties provided that one works with coefficients in a Novikov ring whose degreezero part Λ0 is a field. The key ingredient is a theorem about linear transformations of finitedimensional vector spaces over Novikov fields such as Λ0, which among other things asserts that the range of such a transformation is closed with respect to the ultrametric on Λn 0 induced by the valuation on Λ0. 1.
On the quantum homology algebra of toric Fano manifolds preprint 20 [P2] [PY] [R] [S] [V1] [V2] [V3] [W] Polterovich, L., Hamiltonian loops and Arnold’s principle. Topics in singularity theory
, 1997
"... In this paper we study certain algebraic properties of the quantum homology algebra for the class of symplectic toric Fano manifolds. In particular, we examine the semisimplicity of the quantum homology algebra, and the more general property of containing a field as a direct summand. Our main result ..."
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Cited by 4 (0 self)
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In this paper we study certain algebraic properties of the quantum homology algebra for the class of symplectic toric Fano manifolds. In particular, we examine the semisimplicity of the quantum homology algebra, and the more general property of containing a field as a direct summand. Our main result provides an easilyverified sufficient condition for these properties which is independent of the symplectic form. Moreover, we answer a question of Entov and Polterovich negatively by providing examples of toric Fano manifolds with non semisimple quantum homology algebra. 1 Introduction. The quantum homology algebra QH∗(X,ω) of a symplectic manifold (X,ω) is, roughly speaking, the singular homology of X endowed with a modified algebraic structure, which is a deformation of the ordinary intersection product. It was originally introduced by the string theorists Vafa and Witten [37],[38] in the context of topological quantum field theory,
Displacing Lagrangian toric fibers via probes
"... Abstract. 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 poly ..."
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Cited by 4 (1 self)
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Abstract. 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. 1.
MONODROMY IN HAMILTONIAN FLOER THEORY
, 801
"... Abstract. Schwarz showed that when a closed symplectic manifold (M, ω) is symplectically aspherical (i.e. the symplectic form and the first Chern class vanish on π2(M)) then the spectral invariants, which are initially defined on the universal cover of the Hamiltonian group, descend to the Hamiltoni ..."
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Cited by 2 (1 self)
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Abstract. Schwarz showed that when a closed symplectic manifold (M, ω) is symplectically aspherical (i.e. the symplectic form and the first Chern class vanish on π2(M)) then the spectral invariants, which are initially defined on the universal cover of the Hamiltonian group, descend to the Hamiltonian group Ham(M, ω). 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, ω). For example, they descend if the quantum multiplication of M is undeformed and H2(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, ω) 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.
Quasimorphisms and the Poisson bracket
, 2008
"... For a class of symplectic manifolds, we introduce a functional which assigns a real number to any pair of continuous functions on the manifold. This functional has a number of interesting properties. On the one hand, it is Lipschitz with respect to the uniform norm. On the other hand, it serves as a ..."
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For a class of symplectic manifolds, we introduce a functional which assigns a real number to any pair of continuous functions on the manifold. This functional has a number of interesting properties. On the one hand, it is Lipschitz with respect to the uniform norm. On the other hand, it serves as a measure of noncommutativity of functions in the sense of the Poisson bracket, the operation which involves first derivatives of the functions. Furthermore, the same functional gives rise to a nontrivial lower bound for the error of the simultaneous measurement of a pair of noncommuting Hamiltonians. These results manifest a link between the algebraic structure of the group of Hamiltonian diffeomorphisms and the function theory on a symplectic manifold. The abovementioned functional comes from a special homogeneous quasimorphism on the universal cover of the group, which is rooted in the Floer theory. a Partially supported by E. and J. Bishop Research Fund.