Motivic classes of degeneracy loci and pointed Brill-Noether varieties

TL;DR

This paper develops formulas for motivic Chern and Hirzebruch classes of degeneracy loci, enabling computation of topological invariants of Brill-Noether varieties, thus linking algebraic geometry with topological characteristics.

Contribution

It introduces explicit formulas for motivic classes of degeneracy loci and applies them to compute invariants of pointed Brill-Noether varieties, a novel connection in the field.

Findings

Formulas for motivic Chern and Hirzebruch classes of degeneracy loci

Computed topological invariants of classical and pointed Brill-Noether varieties

Established links between motivic classes and topological characteristics

Abstract

Motivic Chern and Hirzebruch classes are polynomials with K-theory and homology classes as coefficients, which specialize to Chern-Schwartz-MacPherson classes, K-theory classes, and Cappell-Shaneson L-classes. We provide formulas to compute the motivic Chern and Hirzebruch classes of Grassmannian and vexillary degeneracy loci. We apply our results to obtain the Hirzebruch $\chi_y$-genus of classical and one-pointed Brill-Noether varieties, and therefore their topological Euler characteristic, holomorphic Euler characteristic, and signature.