On natural maps from strata of quiver Grassmannians to ordinary Grassmannians

TL;DR

This paper extends the study of quiver Grassmannians from the Kronecker quiver to generalized cases, computing virtual Poincare polynomials of related varieties and revealing new algebraic properties.

Contribution

It introduces a method to compute virtual Poincare polynomials for generalized Kronecker quiver varieties, highlighting differences from the classical case.

Findings

Virtual Poincare polynomials can have negative coefficients.

Explicit formulas for noncommutative cluster variables are utilized.

Generalized quiver Grassmannian projections exhibit distinct geometric properties.

Abstract

Caldero and Zelevinsky studied the geometry of quiver Grassmannians for the Kronecker quiver and computed their Euler characteristics by examining natural stratification of quiver Grassmannians. We consider generalized Kronecker quivers and compute virtual Poincare polynomials of certain varieties which are the images under projections from strata of quiver Grassmannians to ordinary Grassmannians. In contrast to the Kronecker quiver case, these polynomials do not necessarily have positive coefficients. The key ingredient is the explicit formula for noncommutative cluster variables given by Ralf Schiffler and the first author.