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 $$(X, \omega)$$ be any symplectic manifold and


 * $$\mu : X \to \mathbb{R}$$

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

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


 * $$\overline{X}_{\mu \geq \epsilon}$$

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

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

Sometimes it is useful to view the two halves of the symplectic cut as being joined along their shared submanifold $$V$$ to produce a singular space


 * $$\overline{X}_{\mu \leq \epsilon} \cup_V \overline{X}_{\mu \geq \epsilon}.$$

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 $$(X, \omega)$$ be any symplectic manifold. Assume that the circle group $$U(1)$$ acts on $$X$$ in a Hamiltonian way with moment map


 * $$\mu : X \to \mathbb{R}.$$

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


 * $$\omega \oplus (-i dz \wedge d\bar{z}).$$

The group $$U(1)$$ acts on the product in a Hamiltonian way by


 * $$e^{i\theta} \cdot (x, z) = (e^{i \theta} \cdot x, e^{-i \theta} z)$$

with moment map


 * $$\nu(x, z) = \mu(x) - |z|^2.$$

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

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


 * $$X_{> \epsilon} := \mu^{-1}((\epsilon, \infty))$$

and the circle.

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


 * $$\overline{X}_{\mu \geq \epsilon} := \nu^{-1}(\epsilon) / U(1).$$

By construction, it contains $$X_{\mu > \epsilon}$$ as a dense open submanifold; essentially, it compactifies this open manifold with the symplectic quotient


 * $$V := \mu^{-1}(\epsilon) / U(1),$$

which is a symplectic submanifold of $$\overline{X}_{\mu \geq \epsilon}$$ of codimension two.

If $$X$$ is Kähler, then so is the cut space $$\overline{X}_{\mu \geq \epsilon}$$; however, the embedding of $$X_{\mu > \epsilon}$$ is not an isometry.

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

The existence of a global Hamiltonian circle action on $$X$$ 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 $$\mu^{-1}(\epsilon)$$ (since the cut is a local operation).

Blow up as cut
When a complex manifold $$X$$ is blown up along a submanifold $$Z$$, the blow up locus $$Z$$ is replaced by an exceptional divisor $$E$$ and the rest of the manifold is left undisturbed. Topologically, this operation may also be viewed as the removal of an $$\epsilon$$-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 $$(X, \omega)$$ be a symplectic manifold with a Hamiltonian $$U(1)$$-action with moment map $$\mu$$. Assume that the moment map is proper and that it achieves its maximum $$m$$ exactly along a symplectic submanifold $$Z$$ of $$X$$. Assume furthermore that the weights of the isotropy representation of $$U(1)$$ on the normal bundle $$N_X Z$$ are all $$1$$.

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