  Given a smooth curve $C$, we define and study analogues of KLR algebras and
quiver Schur algebras, where quiver representations are replaced by torsion
sheaves on $C$. In particular, they provide a geometric realization for certain
affinized symmetric algebras. When $C=\mathbb P^1$, a version of curve Schur
algebra turns out to be Morita equivalent to the imaginary semi-cuspidal
category of the Kronecker quiver in any characteristic. As a consequence, we
argue that one should not expect to have a reasonable theory of parity sheaves
for affine quivers.
