1996/08/23 by Dan Edidin, Edidin, Dan, W. A. G. GRAHAM +1
Engineering · #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #FOS: Mathematics #Manufacturing Process and Optimization
paper · pdf · doi:10.48550/arxiv.alg-geom/9608028
openalex publication_date 1996/08/23 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
Let X be a projective variety with a torus action, which for simplicity we assume to have dimension 1. If X is a smooth complex variety, then the geometric invariant theory quotient X//G can be identifed with the symplectic reduction Xr. Lerman introduced a construction (valid for symplectic manifolds) called symplectic cutting, which constructs a manifold Xc, such that Xc is the union of Xr and an open subset X>0 ⊂ X. Moreover, there is a natural torus action on Xc such that Xr is a component of the fixed locus. Using localization for equivariant cohomology, this construction can be used to study of Xr. In this note, we give an algebraic version of this construction valid for projective but possibly singular varieties defined over arbitrary fields. This construction is useful for studying Xr from the point of view of algebraic geometry, using the equivariant intersection theory developed by the authors. At the end of the paper we briefly give an adaptation of Lerman's proof of the Kalkman residue formula and use it to give some formulas for characteristic numbers of quotients by a torus.