A Vanishing Result for the Universal Bundle on a Toric Quiver Variety

Abstract

Let $Q$ be a finite quiver without oriented cycles. Denote by $\UB \ra \cwtM$ the fine moduli space of stable thin sincere representations of $Q$ with respect to the canonical stability notion. We prove $\mExt^l_{\cwtM}(\UB,\UB) = 0$ for all $l>0$ and compute the endomorphism algebra of the universal bundle $\UB$. Moreover, we obtain a necessary and sufficient condition for when this algebra is isomorphic to the path algebra of the quiver $Q$. If so, then the bounded derived categories of finitely generated right $\ck Q$-modules and that of coherent sheaves on $\cwtM$ are related via the full and faithful functor $- \otimes^{\Ll}_{\ck Q}\UB$.