Gyroradius

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

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

${\bar{X}}_{\mu \ge ϵ}$

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

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

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

${\bar{X}}_{\mu \le ϵ}{\cup }_V{\bar{X}}_{\mu \ge ϵ}.$

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

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

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

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

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

and the circle.

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

${\bar{X}}_{\mu \ge ϵ}:={\nu }^{-1}(ϵ)/U(1).$

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

$V:={\mu }^{-1}(ϵ)/U(1),$

which is a symplectic submanifold of ${\bar{X}}_{\mu \ge ϵ}$ of codimension two.

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

One constructs ${\bar{X}}_{\mu \le ϵ}$, 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 ${\bar{X}}_{\mu \ge ϵ}$ and ${\bar{X}}_{\mu \le ϵ}$ 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 ${\mu }^{-1}(ϵ)$ (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 $ϵ$-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 $ϵ$ the only critical points in $X_{\mu >m-ϵ}$ are those on $Z$. The symplectic cut ${\bar{X}}_{\mu \le m-ϵ}$, which is formed by deleting a symplectic $ϵ$-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.