Local Euler Obstructions and Sectional Euler Characteristics of Recursive Group Orbits

Xiping Zhang

Published 2020-09-20Version 1

The local Euler obstructions of a projective variety and the Euler characteristics of its linear sections with given hyperplanes are key geometric invariants in the study of singularity theory. Despite their importance, in general it is very hard to compute them. In this paper we consider a special type of singularity: the recursive group orbits. They are the group orbits of a sequence of $G_n$ representations $V_n$ satisfy certain assumptions. We introduce a new intrinsic invariant called the $c_{sm}$ invariant, and use it to give formulas to the local Euler obstructions and sectional Euler characteristics of such orbits. In particular, the matrix rank loci are examples of recursive group orbits. Thus as application, we explicitly compute these geometry invariants for ordinary, skew-symmetric and symmetric rank loci. Our method is algebraic, thus works for algebraically closed field of characteristic $0$.