In mathematics, specifically in symplectic geometry, the symplectic sum is a geometric modification on symplectic manifolds, which glues two given manifolds into a single new one. It is a symplectic version of connected summation along a submanifold, often called a fiber sum.
Contents
- Definition
- Generalizations
- Identity element
- Symplectic sum and cut as deformation
- History and applications
- References
The symplectic sum is the inverse of the symplectic cut, which decomposes a given manifold into two pieces. Together the symplectic sum and cut may be viewed as a deformation of symplectic manifolds, analogous for example to deformation to the normal cone in algebraic geometry.
The symplectic sum has been used to construct previously unknown families of symplectic manifolds, and to derive relationships among the Gromov–Witten invariants of symplectic manifolds.
Definition
Let
such that the Euler classes of the normal bundles are opposite:
In the 1995 paper that defined the symplectic sum, Robert Gompf proved that for any orientation-reversing isomorphism
there is a canonical isotopy class of symplectic structures on the connected sum
meeting several conditions of compatibility with the summands
To produce a well-defined symplectic structure, the connected sum must be performed with special attention paid to the choices of various identifications. Loosely speaking, the isomorphism
Generalizations
In greater generality, the symplectic sum can be performed on a single symplectic manifold
Additionally, the sum can be performed simultaneously on submanifolds
Other generalizations also exist. However, it is not possible to remove the requirement that
A symplectic sum along a submanifold of codimension
But this second cohomology group is zero unless
Identity element
Given
This
So for any particular pair
Symplectic sum and cut as deformation
It is sometimes profitable to view the symplectic sum as a family of manifolds. In this framework, the given data
in which the central fiber is the singular space
obtained by joining the summands
Loosely speaking, one constructs this family as follows. Choose a nonvanishing holomorphic section
Then, in the direct sum
with
for a chosen small
As
An important example occurs when one of the summands is an identity element
However, the symplectic sum is not a complex operation in general. The sum of two Kähler manifolds need not be Kähler.
History and applications
The symplectic sum was first clearly defined in 1995 by Robert Gompf. He used it to demonstrate that any finitely presented group appears as the fundamental group of a symplectic four-manifold. Thus the category of symplectic manifolds was shown to be much larger than the category of Kähler manifolds.
Around the same time, Eugene Lerman proposed the symplectic cut as a generalization of symplectic blow up and used it to study the symplectic quotient and other operations on symplectic manifolds.
A number of researchers have subsequently investigated the behavior of pseudoholomorphic curves under symplectic sums, proving various versions of a symplectic sum formula for Gromov–Witten invariants. Such a formula aids computation by allowing one to decompose a given manifold into simpler pieces, whose Gromov–Witten invariants should be easier to compute. Another approach is to use an identity element
A formula for the Gromov–Witten invariants of a symplectic sum then yields a recursive formula for the Gromov–Witten invariants of