Algebraic Cuts

Abstract

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 $X_r$. Lerman introduced a construction (valid for symplectic manifolds) called symplectic cutting, which constructs a manifold $X_c$, such that $X_c$ is the union of $X_r$ and an open subset $X_{>0} \subset X$. Moreover, there is a natural torus action on $X_c$ such that $X_r$ is a component of the fixed locus. Using localization for equivariant cohomology, this construction can be used to study of $X_r$. 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 $X_r$ 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.