Euler Characteristics of Generic Quiver Grassmannians: Semi-Invariants and Localization

We study the topological Euler characteristic of the quiver Grassmannian for a generic representation. We give a Chern class formula and a finite torus-localization formula. If the quiver $Q$ is acyclic, the formula can be combined with the covariant calculation of Derksen–Schofield–Weyman to obtain a finite integral linear combination of covariant multiplicities, or equivalently of semi-invariant multiplicities for a single flag-extended quiver of $Q$. We also give explicit formulas and examples for generalized Kronecker quivers, and record the same construction for multiplicative characteristic genera.