Modules over plane curve singularities in any ranks and DAHA

Abstract

We generalize the construction of geometric superpolynomials for unibranch plane curve singularities from our prior paper from rank one to any ranks. The new feature is the definition of counterparts of Jacobian factors (directly related to compactified Jacobians) for higher ranks, which is parallel to the classical passage from invertible bundles to vector bundles over algebraic curves. This is an entirely local theory, connected with affine Springer fibers for non-reduced (germs of) spectral curves. We conjecture and justify numerically the connection of our geometric polynomials in arbitrary ranks with the corresponding DAHA superpolynomials of algebraic knots colored by columns.