Allen Knutson

Abstract:   Instead of gluing together simplices along faces, producing a real n-dimensional space, one can use the same data to glue projective spaces along coordinate subspaces, producing a complex n-dimensional variety, whose coordinate ring is the Stanley-Reisner ring of the simplicial complex.

For a wide class of varieties X, I'll define a reverse operation taking X to a simplicial complex; it is reverse in the sense that X has a flat degeneration to the variety associated to the simplicial complex (up to a conjecturally unnecessary "S2ification"). One can then study geometric properties of X in terms of combinatorial properties of the complex, the "clique complex of the Bialynicki-Birula closure relation".

In the case X a flag manifold, this specializes to the geometric interpretation of the Littelmann path model in representation theory, where the closure relation is the Bruhat order. More generally, the theory applies when X has a torus action with isolated fixed points: this includes toric varieties, Springer fibers, and the Hilbert scheme of n points in the projective plane.