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

:$mu\; :\; X\; o\; mathbb\{R\}$

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

:$mu\; :\; X\; o\; mathbb\{R\}.$

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

:

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

:$e^\{i\; heta\}\; cdot\; (x,\; z)\; =\; (e^\{i\; heta\}\; cdot\; x,\; e^\{-i\; heta\}\; z)$

with moment map

:$u(x,\; z)\; =\; mu(x)\; -\; |z|^2.$

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

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

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

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 $Z$.

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.