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

Let $Q$ be a finite acyclic quiver and let $\operatorname{Gr}_\beta(V)$ be a generic quiver Grassmannian arising from a dominant incidence morphism. Combining its regular-zero-locus description with Chern--Gauss--Bonnet and the covariant formula of Derksen--Schofield--Weyman, we express $\chi_{\mathrm{top}}(\operatorname{Gr}_\beta(V))$ as a finite signed sum of covariant multiplicities, equivalently of semi-invariant weight-space dimensions on one flag-extended quiver. We also give a finite torus-localization formula for arbitrary $Q$. We illustrate these results in the cases of generalized Kronecker quivers.