KLR algebras were created by Khovanov and Lauda, and Rouquier as a tool for categorification of quantum groups. Geometric construction of these algebras was constructed by Varagnolo and Vasserot, and Rouquier, and positive characteristic version was constructed by R. Maksimau [J. Algebra 422, 563–610 (2015; Zbl 1387.17030)].
The geometric construction is as follows. Let \(\Gamma\) be a quiver without loops and let \(\alpha\) be the dimension vector. To this data, one can associate a complex variety \(\mathbb{Z}_\alpha\). Its points are parameterized by triples, consisting of a representation of \(\Gamma\) having dimension \(\alpha\) together with two full flags of subrepresentations on it. Then the algebra \(R(\alpha)\) is isomorphic to the equivariant Borel–Moore homology \(H_*^{G_\alpha}(\mathbb{Z}_\alpha)\), where \(G_\alpha\) is a certain group of gauge transformations. The union of categories of (graded, projective, finitely-generated) \(R(\alpha)\)-modules can be then equipped with induction and restriction functors. These functors categorify product and coproduct in the quantum group \(U^-_q(\mathfrak g_\Gamma)\), where \(\mathfrak g_\Gamma\) is the Kac-Moody Lie algebra associated to \(\Gamma\).
Now, given a smooth curve \(C\), the authors define and study analogues of KLR algebras and quiver Schur algebras, where quiver representations are replaced by torsion sheaves on the curve. 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, one should not expect to have a reasonable theory of parity sheaves for affine quivers.