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We study the geometry of equivariant, proper maps from homogeneous bundles$G\times_P V$ over flag varieties $G/P$ to representations of $G$, calledcollapsing maps. Kempf showed that, provided the bundle is completelyreducible, the image $G\cdot V$ of a collapsing map has rational singularitiesin characteristic zero. We extend this result to positive characteristic andshow 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 thatin this case the images of collapsing maps of homogeneous bundles restricted toSchubert varieties are $F$-rational in positive characteristic, and haverational singularities in characteristic zero. We provide results on thesingularities and defining equations of saturations $G\cdot X$ for $P$-stableclosed subvarieties $X\subset V$. We give criteria for the existence of goodfiltrations 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 ofimportant varieties: multicones over Schubert varieties, determinantalvarieties in the space of matrices, symmetric matrices, skew-symmetricmatrices, and certain matrix Schubert varieties therein, representationvarieties of radical square zero algebras (e.g. varieties of complexes),subspace varieties, higher rank varieties, etc.

On the collapsing of homogeneous bundles in arbitrary characteristic