Orbital varieties, and conormal varieties to Schubert varieties

Abstract

Schubert varieties, being the foundational objects of enumerative geometry, have been studied by mathematicians for over a century. Their conormal varieties, and the closely related orbital varieties, have come to play a key role in the representation theory of Lie algebras and algebraic groups. While the combinatorics, and the associated representation theory, of these objects is somewhat well-understood, little is known about their local geometry. In this dissertation, we study primarily the conormal varieties of Schubert varieties in a cominuscule Grassmannian, and the associated orbital varieties. We first show that a compactification given by Lakshmibai, Ravikumar, and Slofstra, identifying the cotangent bundle of the Grassmannian with an open subset of an affine Schubert variety, is the best possible in type A, and does not extend to partial flag varieties. By studying the images of conormal varieties under this compactification, we show that certain conormal varieties can be identified as open subsets of certain affine Schubert varieties, and hence are normal, Cohen-Macaulay, and compatibly Frobenius split. As a corollary, this allows us to compute the projective dual of various determinantal varieties. Next, we mimic the above compactification to show that the conormal varieties of certain Schubert divisors (namely those corresponding to long roots) in a general flag variety can be identified as open subsets of certain affine Schubert varieties. We also show that the same is true for orbital varieties of minimal shape. Finally, we construct a resolution of singularities, and in types A and C, compute a system of defining equations, for the conormal variety of any Schubert subvariety of a cominuscule Grassmannian. As a corollary, this recovers a known system of defining equations for two-column orbital varieties, and also suggests a natural, type-independent, conjecture that could be a first step towards obtaining a system of defining equations in other cominuscule cases.