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 cut was introduced in 1995 by Eugene Lerman, who used it to study the symplectic quotient and other operations on manifolds.
is fibered in circles, each of which is an integral curve of the induced Hamiltonian vector field.
, and one can form a manifold by collapsing each circle fiber to a point.
removed and the boundary collapsed along each circle fiber.
The symplectic cut is the pair of manifolds
Sometimes it is useful to view the two halves of the symplectic cut as being joined along their shared submanifold
to produce a singular space For example, this singular space is the central fiber in the symplectic sum regarded as a deformation.
The preceding description is rather crude; more care is required to keep track of the symplectic structure on the symplectic cut.
in a Hamiltonian way with moment map This moment map can be viewed as a Hamiltonian function that generates the circle action.
, comes with an induced symplectic form The group
acts on the product in a Hamiltonian way by with moment map Let
be any real number such that the circle action is free on
The complement of the submanifold, which consists of points
, is naturally identified with the product of and the circle.
inherits the Hamiltonian circle action, as do its two submanifolds just described.
So one may form the symplectic quotient By construction, it contains
as a dense open submanifold; essentially, it compactifies this open manifold with the symplectic quotient which is a symplectic submanifold of
, the other half of the symplectic cut, in a symmetric manner.
in the two halves of the cut are opposite each other (meaning symplectically anti-isomorphic).
The existence of a global Hamiltonian circle action on
However, it is not actually necessary; the cut can be performed under more general hypotheses, such as a local Hamiltonian circle action near
and the rest of the manifold is left undisturbed.
Topologically, this operation may also be viewed as the removal of an
-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.
Assume that the moment map is proper and that it achieves its maximum
Assume furthermore that the weights of the isotropy representation of
and collapsing the boundary, is then the symplectic blow up of