Isotropic and Coisotropic Subvarieties of Grassmannians

TL;DR

This paper extends the concept of coisotropic hypersurfaces to subvarieties of Grassmannians with arbitrary codimension, exploring their properties and dual isotropic varieties through rank conditions on conormal and tangent spaces.

Contribution

It generalizes coisotropic hypersurfaces to higher codimension subvarieties and introduces the study of isotropic varieties via rank conditions, broadening the theoretical framework.

Findings

Generalization of coisotropic hypersurfaces to arbitrary codimension.

Introduction of isotropic varieties via tangent space conditions.

Connections between coisotropic and isotropic varieties in Grassmannians.

Abstract

We generalize the notion of coisotropic hypersurfaces to subvarieties of Grassmannians having arbitrary codimension. To every projective variety X, Gel'fand, Kapranov and Zelevinsky associate a series of coisotropic hypersurfaces in different Grassmannians. These include the Chow form and the Hurwitz form of X. Gel'fand, Kapranov and Zelevinsky characterized coisotropic hypersurfaces by a rank one condition on conormal spaces, which we use as the starting point for our generalization. We also study the dual notion of isotropic varieties by imposing rank one conditions on tangent spaces instead of conormal spaces.