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.


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