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.
Let ( X , ω ) be any symplectic manifold and
μ : X → R a Hamiltonian on X . Let ϵ be any regular value of μ , so that the level set μ − 1 ( ϵ ) is a smooth manifold. Assume furthermore that μ − 1 ( ϵ ) is fibered in circles, each of which is an integral curve of the induced Hamiltonian vector field.
Under these assumptions, μ − 1 ( [ ϵ , ∞ ) ) is a manifold with boundary μ − 1 ( ϵ ) , and one can form a manifold
X ¯ μ ≥ ϵ by collapsing each circle fiber to a point. In other words, X ¯ μ ≥ ϵ is X with the subset μ − 1 ( ( − ∞ , ϵ ) ) removed and the boundary collapsed along each circle fiber. The quotient of the boundary is a submanifold of X ¯ μ ≥ ϵ of codimension two, denoted V .
Similarly, one may form from μ − 1 ( ( − ∞ , ϵ ] ) a manifold X ¯ μ ≤ ϵ , which also contains a copy of V . The symplectic cut is the pair of manifolds X ¯ μ ≤ ϵ and X ¯ μ ≥ 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
X ¯ μ ≤ ϵ ∪ V X ¯ μ ≥ ϵ . 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. For this, let ( X , ω ) be any symplectic manifold. Assume that the circle group U ( 1 ) acts on X in a Hamiltonian way with moment map
μ : X → R . This moment map can be viewed as a Hamiltonian function that generates the circle action. The product space X × C , with coordinate z on C , comes with an induced symplectic form
ω ⊕ ( − i d z ∧ d z ¯ ) . The group U ( 1 ) acts on the product in a Hamiltonian way by
e i θ ⋅ ( x , z ) = ( e i θ ⋅ x , e − i θ z ) with moment map
ν ( x , z ) = μ ( x ) − | z | 2 . Let ϵ be any real number such that the circle action is free on μ − 1 ( ϵ ) . Then ϵ is a regular value of ν , and ν − 1 ( ϵ ) is a manifold.
This manifold ν − 1 ( ϵ ) contains as a submanifold the set of points ( x , z ) with μ ( x ) = ϵ and | z | 2 = 0 ; this submanifold is naturally identified with μ − 1 ( ϵ ) . The complement of the submanifold, which consists of points ( x , z ) with μ ( x ) > ϵ , is naturally identified with the product of
X > ϵ := μ − 1 ( ( ϵ , ∞ ) ) and the circle.
The manifold ν − 1 ( ϵ ) inherits the Hamiltonian circle action, as do its two submanifolds just described. So one may form the symplectic quotient
X ¯ μ ≥ ϵ := ν − 1 ( ϵ ) / U ( 1 ) . By construction, it contains X μ > ϵ as a dense open submanifold; essentially, it compactifies this open manifold with the symplectic quotient
V := μ − 1 ( ϵ ) / U ( 1 ) , which is a symplectic submanifold of X ¯ μ ≥ ϵ of codimension two.
If X is Kähler, then so is the cut space X ¯ μ ≥ ϵ ; however, the embedding of X μ > ϵ is not an isometry.
One constructs X ¯ μ ≤ 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 X ¯ μ ≥ ϵ and X ¯ μ ≤ 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 μ − 1 ( ϵ ) (since the cut is a local operation).
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 , ω ) be a symplectic manifold with a Hamiltonian U ( 1 ) -action with moment map μ . 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 μ > m − ϵ are those on Z . The symplectic cut X ¯ μ ≤ 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 .
