|Simplicial complexes and algebraic varieties|
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.
|Web page: Alexandru I. Suciu||Comments to: firstname.lastname@example.org|
|Updated: September 10, 2002||URL: http://www.math.neu.edu/bhmn/knutson.html|