Symplectic cut

In mathematics, specifically in symplectic geometry, the symplectic cut is a geometric modification on symplectic manifolds. 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 blow up. The cut was introduced in 1995 by Eugene Lerman, who used it to study the symplectic quotient and other operations on manifolds.

Topological description

Let [math]\displaystyle{ (X, \omega) }[/math] be any symplectic manifold and

[math]\displaystyle{ \mu : X \to \mathbb{R} }[/math]

a Hamiltonian on [math]\displaystyle{ X }[/math]. Let [math]\displaystyle{ \epsilon }[/math] be any regular value of [math]\displaystyle{ \mu }[/math], so that the level set [math]\displaystyle{ \mu^{-1}(\epsilon) }[/math] is a smooth manifold. Assume furthermore that [math]\displaystyle{ \mu^{-1}(\epsilon) }[/math] is fibered in circles, each of which is an integral curve of the induced Hamiltonian vector field.

Under these assumptions, [math]\displaystyle{ \mu^{-1}([\epsilon, \infty)) }[/math] is a manifold with boundary [math]\displaystyle{ \mu^{-1}(\epsilon) }[/math], and one can form a manifold

[math]\displaystyle{ \overline{X}_{\mu \geq \epsilon} }[/math]

by collapsing each circle fiber to a point. In other words, [math]\displaystyle{ \overline{X}_{\mu \geq \epsilon} }[/math] is [math]\displaystyle{ X }[/math] with the subset [math]\displaystyle{ \mu^{-1}((-\infty, \epsilon)) }[/math] removed and the boundary collapsed along each circle fiber. The quotient of the boundary is a submanifold of [math]\displaystyle{ \overline{X}_{\mu \geq \epsilon} }[/math] of codimension two, denoted [math]\displaystyle{ V }[/math].

Similarly, one may form from [math]\displaystyle{ \mu^{-1}((-\infty, \epsilon]) }[/math] a manifold [math]\displaystyle{ \overline{X}_{\mu \leq \epsilon} }[/math], which also contains a copy of [math]\displaystyle{ V }[/math]. The symplectic cut is the pair of manifolds [math]\displaystyle{ \overline{X}_{\mu \leq \epsilon} }[/math] and [math]\displaystyle{ \overline{X}_{\mu \geq \epsilon} }[/math].

Sometimes it is useful to view the two halves of the symplectic cut as being joined along their shared submanifold [math]\displaystyle{ V }[/math] to produce a singular space

[math]\displaystyle{ \overline{X}_{\mu \leq \epsilon} \cup_V \overline{X}_{\mu \geq \epsilon}. }[/math]

For example, this singular space is the central fiber in the symplectic sum regarded as a deformation.

Symplectic description

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]\displaystyle{ (X, \omega) }[/math] be any symplectic manifold. Assume that the circle group [math]\displaystyle{ U(1) }[/math] acts on [math]\displaystyle{ X }[/math] in a Hamiltonian way with moment map

[math]\displaystyle{ \mu : X \to \mathbb{R}. }[/math]

This moment map can be viewed as a Hamiltonian function that generates the circle action. The product space [math]\displaystyle{ X \times \mathbb{C} }[/math], with coordinate [math]\displaystyle{ z }[/math] on [math]\displaystyle{ \mathbb{C} }[/math], comes with an induced symplectic form

[math]\displaystyle{ \omega \oplus (-i dz \wedge d\bar{z}). }[/math]

The group [math]\displaystyle{ U(1) }[/math] acts on the product in a Hamiltonian way by

[math]\displaystyle{ e^{i\theta} \cdot (x, z) = (e^{i \theta} \cdot x, e^{-i \theta} z) }[/math]

with moment map

[math]\displaystyle{ \nu(x, z) = \mu(x) - |z|^2. }[/math]

Let [math]\displaystyle{ \epsilon }[/math] be any real number such that the circle action is free on [math]\displaystyle{ \mu^{-1}(\epsilon) }[/math]. Then [math]\displaystyle{ \epsilon }[/math] is a regular value of [math]\displaystyle{ \nu }[/math], and [math]\displaystyle{ \nu^{-1}(\epsilon) }[/math] is a manifold.

This manifold [math]\displaystyle{ \nu^{-1}(\epsilon) }[/math] contains as a submanifold the set of points [math]\displaystyle{ (x, z) }[/math] with [math]\displaystyle{ \mu(x) = \epsilon }[/math] and [math]\displaystyle{ |z|^2 = 0 }[/math]; this submanifold is naturally identified with [math]\displaystyle{ \mu^{-1}(\epsilon) }[/math]. The complement of the submanifold, which consists of points [math]\displaystyle{ (x, z) }[/math] with [math]\displaystyle{ \mu(x) \gt \epsilon }[/math], is naturally identified with the product of

[math]\displaystyle{ X_{\gt \epsilon} := \mu^{-1}((\epsilon, \infty)) }[/math]

and the circle.

The manifold [math]\displaystyle{ \nu^{-1}(\epsilon) }[/math] inherits the Hamiltonian circle action, as do its two submanifolds just described. So one may form the symplectic quotient

[math]\displaystyle{ \overline{X}_{\mu \geq \epsilon} := \nu^{-1}(\epsilon) / U(1). }[/math]

By construction, it contains [math]\displaystyle{ X_{\mu \gt \epsilon} }[/math] as a dense open submanifold; essentially, it compactifies this open manifold with the symplectic quotient

[math]\displaystyle{ V := \mu^{-1}(\epsilon) / U(1), }[/math]

which is a symplectic submanifold of [math]\displaystyle{ \overline{X}_{\mu \geq \epsilon} }[/math] of codimension two.

If [math]\displaystyle{ X }[/math] is Kähler, then so is the cut space [math]\displaystyle{ \overline{X}_{\mu \geq \epsilon} }[/math]; however, the embedding of [math]\displaystyle{ X_{\mu \gt \epsilon} }[/math] is not an isometry.

One constructs [math]\displaystyle{ \overline{X}_{\mu \leq \epsilon} }[/math], the other half of the symplectic cut, in a symmetric manner. The normal bundles of [math]\displaystyle{ V }[/math] in the two halves of the cut are opposite each other (meaning symplectically anti-isomorphic). The symplectic sum of [math]\displaystyle{ \overline{X}_{\mu \geq \epsilon} }[/math] and [math]\displaystyle{ \overline{X}_{\mu \leq \epsilon} }[/math] along [math]\displaystyle{ V }[/math] recovers [math]\displaystyle{ X }[/math].

The existence of a global Hamiltonian circle action on [math]\displaystyle{ 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]\displaystyle{ \mu^{-1}(\epsilon) }[/math] (since the cut is a local operation).

Blow up as cut

When a complex manifold [math]\displaystyle{ X }[/math] is blown up along a submanifold [math]\displaystyle{ Z }[/math], the blow up locus [math]\displaystyle{ Z }[/math] is replaced by an exceptional divisor [math]\displaystyle{ E }[/math] and the rest of the manifold is left undisturbed. Topologically, this operation may also be viewed as the removal of an [math]\displaystyle{ \epsilon }[/math]-neighborhood of the blow up locus, followed by the collapse of the boundary by the Hopf map.

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.

As before, let [math]\displaystyle{ (X, \omega) }[/math] be a symplectic manifold with a Hamiltonian [math]\displaystyle{ U(1) }[/math]-action with moment map [math]\displaystyle{ \mu }[/math]. Assume that the moment map is proper and that it achieves its maximum [math]\displaystyle{ m }[/math] exactly along a symplectic submanifold [math]\displaystyle{ Z }[/math] of [math]\displaystyle{ X }[/math]. Assume furthermore that the weights of the isotropy representation of [math]\displaystyle{ U(1) }[/math] on the normal bundle [math]\displaystyle{ N_X Z }[/math] are all [math]\displaystyle{ 1 }[/math].

Then for small [math]\displaystyle{ \epsilon }[/math] the only critical points in [math]\displaystyle{ X_{\mu \gt m - \epsilon} }[/math] are those on [math]\displaystyle{ Z }[/math]. The symplectic cut [math]\displaystyle{ \overline{X}_{\mu \leq m - \epsilon} }[/math], which is formed by deleting a symplectic [math]\displaystyle{ \epsilon }[/math]-neighborhood of [math]\displaystyle{ Z }[/math] and collapsing the boundary, is then the symplectic blow up of [math]\displaystyle{ X }[/math] along [math]\displaystyle{ Z }[/math].

References

• Eugene Lerman: Symplectic cuts, Mathematical Research Letters 2 (1995), 247–258
• Dusa McDuff and D. Salamon: Introduction to Symplectic Topology (1998) Oxford Mathematical Monographs, ISBN:0-19-850451-9.