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| In [[mathematics]], specifically in [[symplectic geometry]], the '''symplectic cut''' is a geometric modification on [[symplectic manifold]]s. Its effect is to decompose a given manifold into two pieces. There is an inverse operation, the [[symplectic sum]], that glues two manifolds together into one. The symplectic cut can also be viewed as a generalization of symplectic [[Blowing up|blow up]]. The cut was introduced in 1995 by Eugene Lerman, who used it to study the [[moment map|symplectic quotient]] and other operations on manifolds.
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| == Topological description ==
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| Let <math>(X, \omega)</math> be any symplectic manifold and
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| :<math>\mu : X \to \mathbb{R}</math>
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| a [[Hamiltonian vector field|Hamiltonian]] on <math>X</math>. Let <math>\epsilon</math> be any regular value of <math>\mu</math>, so that the level set <math>\mu^{-1}(\epsilon)</math> is a smooth manifold. Assume furthermore that <math>\mu^{-1}(\epsilon)</math> is fibered in circles, each of which is an integral curve of the induced [[Hamiltonian vector field]].
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| Under these assumptions, <math>\mu^{-1}([\epsilon, \infty))</math> is a manifold with boundary <math>\mu^{-1}(\epsilon)</math>, and one can form a manifold
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| :<math>\overline{X}_{\mu \geq \epsilon}</math>
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| by collapsing each circle fiber to a point. In other words, <math>\overline{X}_{\mu \geq \epsilon}</math> is <math>X</math> with the subset <math>\mu^{-1}((-\infty, \epsilon))</math> removed and the boundary collapsed along each circle fiber. The quotient of the boundary is a submanifold of <math>\overline{X}_{\mu \geq \epsilon}</math> of [[codimension]] two, denoted <math>V</math>.
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| Similarly, one may form from <math>\mu^{-1}((-\infty, \epsilon])</math> a manifold <math>\overline{X}_{\mu \leq \epsilon}</math>, which also contains a copy of <math>V</math>. The '''symplectic cut''' is the pair of manifolds <math>\overline{X}_{\mu \leq \epsilon}</math> and <math>\overline{X}_{\mu \geq \epsilon}</math>.
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| Sometimes it is useful to view the two halves of the symplectic cut as being joined along their shared submanifold <math>V</math> to produce a singular space
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| :<math>\overline{X}_{\mu \leq \epsilon} \cup_V \overline{X}_{\mu \geq \epsilon}.</math>
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| For example, this singular space is the central fiber in the symplectic sum regarded as a deformation.
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| == Symplectic description ==
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| The preceding description is rather crude; more care is required to keep track of the symplectic structure on the symplectic cut. For this, let <math>(X, \omega)</math> be any symplectic manifold. Assume that the [[circle group]] <math>U(1)</math> [[group action|acts]] on <math>X</math> in a [[moment map|Hamiltonian]] way with [[moment map]]
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| :<math>\mu : X \to \mathbb{R}.</math>
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| This moment map can be viewed as a Hamiltonian function that generates the circle action. The product space <math>X \times \mathbb{C}</math>, with coordinate <math>z</math> on <math>\mathbb{C}</math>, comes with an induced symplectic form
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| :<math>\omega \oplus (-i dz \wedge d\bar{z}).</math>
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| The group <math>U(1)</math> acts on the product in a Hamiltonian way by
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| :<math>e^{i\theta} \cdot (x, z) = (e^{i \theta} \cdot x, e^{-i \theta} z)</math>
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| with moment map
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| :<math>\nu(x, z) = \mu(x) - |z|^2.</math>
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| Let <math>\epsilon</math> be any real number such that the circle action is free on <math>\mu^{-1}(\epsilon)</math>. Then <math>\epsilon</math> is a regular value of <math>\nu</math>, and <math>\nu^{-1}(\epsilon)</math> is a manifold.
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| This manifold <math>\nu^{-1}(\epsilon)</math> contains as a submanifold the set of points <math>(x, z)</math> with <math>\mu(x) = \epsilon</math> and <math>|z|^2 = 0</math>; this submanifold is naturally identified with <math>\mu^{-1}(\epsilon)</math>. The complement of the submanifold, which consists of points <math>(x, z)</math> with <math>\mu(x) > \epsilon</math>, is naturally identified with the product of
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| :<math>X_{> \epsilon} := \mu^{-1}((\epsilon, \infty))</math>
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| and the circle.
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| The manifold <math>\nu^{-1}(\epsilon)</math> inherits the Hamiltonian circle action, as do its two submanifolds just described. So one may form the symplectic quotient
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| :<math>\overline{X}_{\mu \geq \epsilon} := \nu^{-1}(\epsilon) / U(1).</math>
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| By construction, it contains <math>X_{\mu > \epsilon}</math> as a dense open submanifold; essentially, it compactifies this open manifold with the symplectic quotient
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| :<math>V := \mu^{-1}(\epsilon) / U(1),</math> | |
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| which is a symplectic submanifold of <math>\overline{X}_{\mu \geq \epsilon}</math> of codimension two.
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| If <math>X</math> is [[Kähler manifold|Kähler]], then so is the cut space <math>\overline{X}_{\mu \geq \epsilon}</math>; however, the embedding of <math>X_{\mu > \epsilon}</math> is not an isometry.
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| One constructs <math>\overline{X}_{\mu \leq \epsilon}</math>, the other half of the symplectic cut, in a symmetric manner. The [[normal bundle]]s of <math>V</math> in the two halves of the cut are opposite each other (meaning symplectically anti-isomorphic). The symplectic sum of <math>\overline{X}_{\mu \geq \epsilon}</math> and <math>\overline{X}_{\mu \leq \epsilon}</math> along <math>V</math> recovers <math>X</math>.
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| The existence of a global Hamiltonian circle action on <math>X</math> appears to be a restrictive assumption. However, it is not actually necessary; the cut can be performed under more general hypotheses, such as a local Hamiltonian circle action near <math>\mu^{-1}(\epsilon)</math> (since the cut is a local operation).
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| == Blow up as cut ==
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| When a [[complex manifold]] <math>X</math> is blown up along a submanifold <math>Z</math>, the blow up [[locus (mathematics)|locus]] <math>Z</math> is replaced by an [[exceptional divisor]] <math>E</math> and the rest of the manifold is left undisturbed. Topologically, this operation may also be viewed as the removal of an <math>\epsilon</math>-neighborhood of the blow up locus, followed by the collapse of the boundary by the [[Hopf fibration|Hopf map]].
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| Blowing up a symplectic manifold is more subtle, since the symplectic form must be adjusted in a neighborhood of the blow up locus in order to continue smoothly across the exceptional divisor in the blow up. The symplectic cut is an elegant means of making the neighborhood-deletion/boundary-collapse process symplectically rigorous.
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| As before, let <math>(X, \omega)</math> be a symplectic manifold with a Hamiltonian <math>U(1)</math>-action with moment map <math>\mu</math>. Assume that the moment map is proper and that it achieves its maximum <math>m</math> exactly along a symplectic submanifold <math>Z</math> of <math>X</math>. Assume furthermore that the weights of the isotropy representation of <math>U(1)</math> on the normal bundle <math>N_X Z</math> are all <math>1</math>.
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| Then for small <math>\epsilon</math> the only critical points in <math>X_{\mu > m - \epsilon}</math> are those on <math>Z</math>. The symplectic cut <math>\overline{X}_{\mu \leq m - \epsilon}</math>, which is formed by deleting a symplectic <math>\epsilon</math>-neighborhood of <math>Z</math> and collapsing the boundary, is then the symplectic blow up of <math>X</math> along <math>Z</math>.
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| == References ==
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| * Eugene Lerman: Symplectic cuts, ''Mathematical Research Letters'' 2 (1995), 247–258
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| * [[Dusa McDuff]] and D. Salamon: ''Introduction to Symplectic Topology'' (1998) Oxford Mathematical Monographs, ISBN 0-19-850451-9.
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| [[Category:Symplectic topology]]
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