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

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

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

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

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

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

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

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

${\displaystyle {\overline {X}}_{\mu \leq \epsilon }\cup _{V}{\overline {X}}_{\mu \geq \epsilon }.}$

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

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

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

${\displaystyle \omega \oplus (-idz\wedge d{\bar {z}}).}$

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

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

with moment map

${\displaystyle \nu (x,z)=\mu (x)-|z|^{2}.}$

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

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

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

and the circle.

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

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

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

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

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

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

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

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

Blow up as cut

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

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

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.