Combinatorial K-theory

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

The investigator will study the cohomology theory and K-theory of partial flag varieties and quiver varieties. In particular he will look for K-theory parallels of known results in cohomology. A first target is the cohomology and Grothendieck rings of a partial flag variety. Since any partial flag variety has a cell decomposition into Schubert cells, its cohomology ring has a basis consisting of the cohomology classes of Schubert varieties. Likewise the Grothendieck ring of algebraic vector bundles has a basis of structure sheaves of Schubert varieties. The investigator will study the structure constants for these rings with respect to their bases indexed by Schubert varieties. The main goal is to find explicit formula for these constants, but also positivity questions are of interest. In cohomology the structure constants are known to be positive for geometric reasons, but no combinatorial proof of this fact is known. In K-theory the investigator has conjectured that the structure constants have alternating signs, i.e. they are non-negative in even degrees and non-positive in odd degrees. Finding a proof of this would be very interesting. An additional goal is to find efficient computer algorithms for calculating these structure constants. The investigator will also try to find a formula for the K-theory class of the structure sheaf of a general quiver variety. Such a formula will generalize a formula for the cohomology class of a quiver variety, which the investigator has proved earlier with Fulton. The investigator hopes that proving such a formula will be of help for constructing an explicit resolution of the structure sheaf of a quiver variety. This would generalize classical constructions such as the Koszul complex, which is of fundamental importance in homological algebra. The development of cohomology theory was motivated in part by the problem of classifying topological spaces. This is important for addressing questions such as "what is the shape of our universe?". Cohomology theory is also an important tool for solving problems in enumerative geometry, in which one seeks to determine and count all the solutions to a geometric problem. For example, if the geometric problem is to find lines which intersect or are tangent to a given collection of fixed geometric figures, then the number of such lines is desired. A very powerful technique for solving this type of problems is to construct a space consisting of all objects which could potentially be a solution. Counting the number of solutions to a problem is often equivalent to performing a calculation in the cohomology ring of this space of potential solutions. The objects to be counted can in many situations be identified with flags of subspaces in a given vector space. Flag varieties, whose points correspond to such flags of subspaces, are therefore typical candidates to act as the space of potential solutions. This makes it important to be able to do efficient computations within the cohomology ring of a flag variety, and gives the reason why the structure constants for this ring has been wanted by geometers and combinatorialists for decades. The K-theory or Grothendieck ring of a variety can be seen as a generalization of the cohomology ring. A good understanding of K-theory will therefore give a more complete picture of cohomology theory. At the same time K-theory is important for the study of vector bundles on a variety and for homological algebra. This makes it very natural to try to generalize the known results about cohomology theory to K-theory.