On the collapsing of homogeneous bundles in arbitrary characteristic

TL;DR

This paper extends the understanding of the singularities of collapsing maps from homogeneous bundles over flag varieties to representations of algebraic groups, providing a characteristic-free approach applicable in both zero and positive characteristic.

Contribution

It generalizes Kempf's results to positive characteristic, establishing conditions for strong F-regularity and rational singularities of the images of collapsing maps.

Findings

Saturation of collapsing maps is strongly F-regular in positive characteristic under good filtration conditions.

Images of collapsing maps restricted to Schubert varieties are F-rational in positive characteristic.

Provides criteria for good filtrations and a uniform approach for various important algebraic varieties.

Abstract

We study the geometry of equivariant, proper maps from homogeneous bundles $G\times_P V$ over flag varieties $G/P$ to representations of $G$, called collapsing maps. Kempf showed that, provided the bundle is completely reducible, the image $G\cdot V$ of a collapsing map has rational singularities in characteristic zero. We extend this result to positive characteristic and show that for the analogous bundles the saturation $G\cdot V$ is strongly $F$-regular if its coordinate ring has a good filtration. We further show that in this case the images of collapsing maps of homogeneous bundles restricted to Schubert varieties are $F$-rational in positive characteristic, and have rational singularities in characteristic zero. We provide results on the singularities and defining equations of saturations $G\cdot X$ for $P$-stable closed subvarieties $X\subset V$. We give criteria for the existence of good filtrations for the coordinate ring of $G\cdot X$. Our results give a uniform, characteristic-free approach for the study of the geometry of a number of important varieties: multicones over Schubert varieties, determinantal varieties in the space of matrices, symmetric matrices, skew-symmetric matrices, and certain matrix Schubert varieties therein, representation varieties of radical square zero algebras (e.g. varieties of complexes), subspace varieties, higher rank varieties, etc.