This proposal concerns research on some problems in algebraic geometry that arise in the context of group actions and vector bundles. There are four parts to the proposal. The first part of the proposal is to study lef vector bundles on algebraic varieties. In particular the proposer plans to study a conjecture he has made concerning such bundles, and also to extend to lef bundles some positivity results proved by Fulton and Lazarsfeld for ample bundles. The second part of the proposal concerns labeled directed graphs modeled on the moment graphs defined for certain torus actions on algebraic varieties. The ``intersection homology'' Poincare polynomial of such a generalized moment graph is defined, and the goal is to see if this polynomial is palindromic and unimodal for more general graphs of this type, not necessarily arising as moment graphs, but with (as far as can be determined) all the graph-theoretic properties that moment graphs have. The third part of the proposal is to prove a non-negativity theorem about the multiplication in the torus-equivariant K-theory of the flag variety. The fourth part of the proposal is to describe the Riemann-Roch map of a geometric quotient by an algebraic group in terms of the equivariant Riemann-Roch map.
Algebraic geometry is a branch of mathematics that has grown out of the study of solutions to polynomial equations. By linking algebra and geometry, it sheds light on both, and has implications for many other branches of mathematics, including number theory, cryptography, and group theory. This project is largely devoted to studying some problems in algebraic geometry that are related to algebraic groups. To say that a problem is related to algebraic groups means that the problem has some symmetry which can make it more accessible to investigation. Sometimes the symmetry can be used to extract essential features of the problem and study them without considering all of the geometric complexities; the moment graphs, which are part of this project, are an example of this.
