Torus Actions on Symplectic Manifolds

Audin, Michèle

doi:10.1007/978-3-0348-7960-6

Cited by 189 publications

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“…On the other hand, the flat metric on an elliptic curve and the hyperbolic metric on a Riemann surface of genus ≥ 2 cannot be projectively induced (see [27] for a proof) and hence M is forced to be biholomorphic to CP 1 (and g B isometric to an integer multiple of the Fubini-Study metric). (1) mg is not projectively induced for all m; (2) there is at least a non-constant coefficient a j 0 , with j 0 ≥ 2, of the TYZ expansion (5) of the Kempf distortion function T mg (x).…”

Section: Corollary 24 Let (M L) Be a Polarized Manifold And M Have mentioning

confidence: 97%

On homothetic balanced metrics

2011

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In this article, we study the set of balanced metrics given in Donaldson's terminology (J. Diff. Geometry 59:479-522, 2001) on a compact complex manifold M which are homothetic to a given balanced one. This question is related to various properties of the Tian-Yau-Zelditch approximation theorem for Kähler metrics. We prove that this set is finite when M admits a non-positive Kähler-Einstein metric, in the case of non-homogenous toric Kähler-Einstein manifolds of dimension ≤ 4 and in the case of the constant scalar curvature metrics found in Arezzo and Pacard (Acta. Math. 196(2):179-228, 2006; Ann. Math. 170(2):685-738, 2009). © 2011 Springer Science+Business Media B.V

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Section: Corollary 24 Let (M L) Be a Polarized Manifold And M Have mentioning

confidence: 97%

On homothetic balanced metrics

2011

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“…The classification of closed symplectic 4-manifolds with a Hamiltonian SO(3)-or SU(2)-action was given in [1] and [7]. In the rest of the article, a proof to the theorem will be given below, which describes the classification of five-dimensional contact SO(3)-manifolds.…”

Section: Five-dimensional Contact So(3)-manifoldsmentioning

confidence: 98%

“…We can cut R/S 1 open to obtain a disk, and the extra contributions from the cuts cancel out. With the equations ι(γ j , ∂σ 2 ) − ι(γ j , ∂σ 1 ) = rot( f | ∂R j ) for j k, and ι(γ j , ∂σ 2 ) − ι(γ j , ∂σ 1 …”

Section: (T R) R(t R) ϕ + ρ ε (R) · (T R)mentioning

confidence: 99%

Five-Dimensional Contact SU(2)- and SO(3)-Manifolds and Dehn Twists

Niederkrüger¹

2006

Geom Dedicata

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In the first part of this paper the five-dimensional contact SO(3)-manifolds are classified up to equivariant coorientation preserving contactomorphisms. The construction of such manifolds with singular orbits requires the use of generalized Dehn twists. We show as an application that all simply connected 5-manifolds with singular orbits are realized by a Brieskorn manifold with exponents (k, 2, 2, 2). The standard contact structure on such a manifold gives right-handed Dehn twists, and a second contact structure defined in the article gives left-handed twists. In an appendix we also describe the classification of five-dimensional contact SU(2)-manifolds. Mathematics Subject Classifications (2000). 53D35, 53D20.A five-dimensional contact SO(3)-manifold M can be decomposed into the set of singular orbits M (sing) and the set of regular orbits M (reg) . Both parts can be described relatively easily: The singular orbits are the disjoint union of copies of S 1 × RP 2 , S 1 × S 2 or S 1 ×S 2 := R × S 2 /∼, where (t, p) ∼ (t + 1, −p). The set of regular orbits contains a canonical submanifold R of dimension 3 (the so-called crosssection), and one has that M (reg) ∼ = SO(3) × S 1 R.For gluing the singular orbits onto the regular ones, there is an integer invariant that classifies all possibilities. This integer corresponds to the number of Dehn twists.There is a similar decomposition of five-dimensional contact SU(2)-manifolds. Here all singular orbits are fixed points, the components of M (sing) are embedded circles, and the set M (reg) can be described by SU(2) × S 1 R. In contrast to SO(3)-manifolds, there is no freedom in gluing M (reg) to M (sing) .

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“…We fix notation for toric manifolds. For background materials on toric manifolds, we refer the reader to [Oda88,Aud04,CLS11]. Let N ∼ = Z D denote a lattice.…”

Section: Toric Mirror Theoremmentioning

confidence: 99%

Shift operators and toric mirror theorem

Iritani

2017

Geom. Topol.

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Abstract. We give a new proof of Givental's mirror theorem for toric manifolds using shift operators of equivariant parameters. The proof is almost tautological: it gives an A-model construction of the I-function and the mirror map. It also works for non-compact or non-semipositive toric manifolds. IntroductionIn 1995, Seidel [Sei97] introduced an invertible element of quantum cohomology associated to a Hamiltonian circle action. This has had many applications in symplectic topology. Seidel himself used it to construct non-trivial elements of π 1 of the group of Hamiltonian diffeomorphisms. calculated Seidel's elements in a more general setting and obtained Batyrev's ring presentation of quantum cohomology of toric manifolds. Their method, however, does not yield explicit structure constants of quantum cohomology, i.e. genus-zero Gromov-Witten invariants.Recently, Braverman, Maulik, Okounkov and Pandharipande [OP10, BMO11, MO12] introduced a shift operator of equivariant parameters for equivariant quantum cohomology. Their shift operators reduce to Seidel's invertible elements under the nonequivariant limit. In this paper, we show that equivariant genus-zero Gromov-Witten invariants of toric manifolds are reconstructed only from formal properties of shift operators. This means that the equivariant quantum topology of toric manifolds is determined by its classical counterpart.More specifically, we give a new proof of Givental's mirror theorem for toric manifolds, which is stated as follows: Giv98b,LLY99,Iri08,Bro09], see §4.2 for more details). Let X Σ be a semi-projective toric manifold having a torus fixed point. Let I(y, z) be the cohomologyvalued hypergeometric series defined bywhere u i , i = 1, . . . , m is the class of a prime toric divisor. Then I(y, −z) lies in Givental's Lagrangian cone L X Σ associated to X Σ .We prove this theorem in the following way. Recall that equivariant genus-zero Gromov-Witten invariants of a T -variety X can be encoded by an infinite-dimensional Lagrangian submanifold L X of the symplectic vector space H X [Giv04]:The space H X is called the Givental space and L X is called the Givental cone. By the general theory, each C × -subgroup k : C × → T defines a shift operator S k acting on the Givental space H X and induces a vector field on L X :The operator S k is determined by T -fixed loci in X and their normal bundles (see Definition 3.13). For toric manifolds, we have a shift operator S i for each torus-invariant prime divisor. Then we identify the I-function I(y, z) with an integral curve of the commuting vector fields f → z −1 S i f. Theorem 1.2. Givental's I-function I(y, z) is a unique integral curve which satisfies the differential equation:and is of the form I(y, z) = ze, where we setThe I-function defines a mirror map y → τ (y) ∈ H * T (X) via Birkhoff factorization [CG07, Iri08]. As a corollary to our proof, we obtain the following relationship between the equivariant Seidel elements S i (τ ) and the mirror map. This generalizes a previous result [GI12] in the semipo...

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Torus Actions on Symplectic Manifolds

Cited by 189 publications

References 0 publications

On homothetic balanced metrics

On homothetic balanced metrics

Five-Dimensional Contact SU(2)- and SO(3)-Manifolds and Dehn Twists

Shift operators and toric mirror theorem

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