Local Euler obstructions and Chern–Mather classes of determinantal varieties

Abstract

The local Euler obstructions are key ingredients in the study of both singularity theory and geometric representation theory. In this note, we consider matrix rank stratification over algebraically closed field of arbitrary characteristic. We compute the local Euler obstructions using a direct intersection-theoretic approach. The algebraic formula we prove are based on explicitly calculating the integrations of certain Chern classes of the universal bundles over the Grassmannians, and we proceed such calculations using formal Chern roots argument. Our formula generalizes the result over $\mathbb{C}$ computed in [8] via topological methods. Based on our integration formula we propose an explanation to the Pascal triangle pattern observed by the authors in [8]. Then, we generalize the formula of the Chern–Mather class of ${\tau }_{m,n,k}$ in [18, theorem 10] to arbitrary algebraically closed base field. In particular, when k = 1 and $k=n-1$ we explicitly compute the Chern–Mather classes and prove some interesting symmetric patterns on the coefficients.

Keywords:

• Characteristic class
• determinantal variety
• local Euler obstruction

2020 Mathematics Subject Classification:

• 14C17
• 14J17

Acknowledgments

The author would like to thank Paolo Aluffi for all the help and support, and Terence Gaffney and Nivaldo G. Grulha Jr. for the helpful discussions. The author also would like to thank the referees for all the comments.

Additional information

Funding

The author is supported by China Postdoctoral Science Foundation (Grant No.2019M661329).