Chern class formulas for $G_{2}$ Schubert loci

Abstract

We define degeneracy loci for vector bundles with structure group $G_2$ and give formulas for their cohomology (or Chow) classes in terms of the Chern classes of the bundles involved. When the base is a point, such formulas are part of the theory for rational homogeneous spaces developed by Bernstein–Gelfand–Gelfand and Demazure. This has been extended to the setting of general algebraic geometry by Giambelli–Thom–Porteous, Kempf–Laksov, and Fulton in classical types; the present work carries out the analogous program in type $G_2$. We include explicit descriptions of the $G_2$ flag variety and its Schubert varieties, and several computations, including one that answers a question of W. Graham. In appendices, we collect some facts from representation theory and compute the Chow rings of quadric bundles, correcting an error in a paper by Edidin and Graham.