Computing the Chern–Schwartz–MacPherson Class of Complete Simplical Toric Varieties

Abstract

Topological invariants such as characteristic classes are an important tool to aid in understanding and categorizing the structure and properties of algebraic varieties. In this note we consider the problem of computing a particular characteristic class, the Chern-Schwartz-MacPherson class, of a complete simplicial toric variety X Σ defined by a fan Σ from the combinatorial data contained in the fan Σ. Specifically, we give an effective combinatorial algorithm to compute the Chern-Schwartz-MacPherson class of X Σ , in the Chow ring (or rational Chow ring) of X Σ . This method is formulated by combining, and when necessary modifying, several known results from the literature and is implemented in Macaulay2 for test purposes.