Schubert calculus and singularity theory

TL;DR

This paper explores a universal deformation framework for Schubert calculus, connecting classical and generalized cohomology theories through Jacobi rings and singularity theory, offering new insights into their structural relationships.

Contribution

It demonstrates that the cohomology of hermitian symmetric spaces and their deformations can be described as Jacobi rings of specific potentials, unifying various cohomology theories.

Findings

Cohomology of hermitian symmetric spaces is a Jacobi ring of a certain potential.

Equivariant, quantum cohomology, and K-theory are deformations of this potential.

Provides a new perspective on Schubert calculus via singularity theory.

Abstract

Schubert calculus has been in the intersection of several fast developing areas of mathematics for a long time. Originally invented as the description of the cohomology of homogeneous spaces it has to be redesigned when applied to other generalized cohomology theories such as the equivariant, the quantum cohomology, K-theory, and cobordism. All this cohomology theories are different deformations of the ordinary cohomology. In this note we show that there is in some sense the universal deformation of Schubert calculus which produces the above mentioned by specialization of the appropriate parameters. We build on the work of Lerche Vafa and Warner. The main conjecture there was that the classical cohomology of a hermitian symmetric homogeneous manifold is a Jacobi ring of an appropriate potential. We extend this conjecture and provide a simple proof. Namely we show that the cohomology of the hermitian symmetric space is a Jacobi ring of a certain potential and the equivariant and the quantum cohomology and K-theory is a Jacobi ring of a particular deformation of this potential. This suggests to study the most general deformations of the Frobenius algebra of cohomology of these manifolds by considering the versal deformation of the appropriate potential. The structure of the Jacobi ring of such potential is a subject of well developed singularity theory. This gives a potentially new way to look at the classical, the equivariant, the quantum and other flavors of Schubert calculus.