Free resolutions of some Schubert singularities (Q903167)

The authors construct free resolutions for the coordinate rings of some subvarieties of Grassmannians \(G(n,N)\), over the complex field. Considering \(G(n,N)\) as the quotient of \(\mathrm{GL}_N\) by the action of a parabolic group \(P\), then its Schubert subvarieties \(X(w)\) correspond to orbits of (the classes of) elements \(w\in\mathrm{GL}_N\) under the action of the group of upper triangular matrices. Define the \textit{opposite big cell} \(O^-\) as the orbit of the identity under the group of lower triangular matrices. The authors consider \textit{opposite cell varieties}, defined as intersections \(Y(w)=X(w)\cap O^-\). The authors determine a free resolution for opposite cell varieties \(Y(w)\). The construction is effective when \(w\) is suitably chosen in \(\mathrm{GL}_N\). Opposite cell varieties for which the construction is effective fill a wide class of subvarieties of Grassmannians, which properly includes determinantal varieties. The method relies in the computation of the cohomology of a vector bundle over a desingularization of \(Y(w)\). The bundle is a restriction of a homogeneous bundle over a quotient \(P/\tilde P\), for a suitable subgroup \(\tilde P\). The bundle is not completely reducible, yet the authors are able to compute its cohomology, by restricting to the pieces of a suitable filtration of \(P/\tilde P\).