ON THE COLLAPSING OF HOMOGENEOUS BUNDLES IN ARBITRARY CHARACTERISTIC

We study the geometry of equivariant, proper maps from homogeneous bundles G x (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 center dot 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 center dot 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 center dot X for P-stable closed subvarieties X subset of V. We give criteria for the existence of good filtrations for the coordinate ring of G center dot 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.