Alexandre Minets

Abstract: We compute the cohomological Hall algebra of zero-dimensional sheaves on an arbitrary smooth quasi-projective surface \(S\) with pure cohomology, deriving an explicit presentation by generators and relations. When \(S\) has trivial canonical bundle, this COHA is isomorphic to the enveloping algebra of deformed trigonometric \(W_{1+\infty}\)-algebra associated to the ring \(H^*(S,\mathbb{Q}\). We also define a double of this COHA, show that it acts on the homology of various moduli stacks of sheaves on \(S\) and explicitly describe this action on the products of tautological classes. Examples include Hilbert schemes of points on surfaces, the moduli stack of Higgs bundles on a smooth projective curve and the moduli stack of \(1\)-dimensional sheaves on a \(K3\) surface in an ample class. The double COHA is shown to contain Nakajima's Heisenberg algebra, as well as a copy of the Virasoro algebra.

Abstract: Let \(\mathcal{H}_2\) be the Lie algebra of polynomial Hamiltonian vector fields on the symplectic plane. Let \(X\) be the moduli space of stable Higgs bundles of fixed relatively prime rank and degree, or more generally the moduli space of stable parabolic Higgs bundles of arbitrary rank and degree for a generic stability condition. Let \(H^*(X)\) be the cohomology with complex coefficients. Using the operations of cup-product by tautological classes and Hecke correspondences we construct an action of \(\mathcal{H}_2\) on \(H^*(X)[x,y]\), where \(x\) and \(y\) are formal variables. We show that the perverse filtration on \(H^*(X)\) coincides with the filtration canonically associated to \(\mathfrak{sl}_2\subset\mathcal{H}_2\) and deduce the \(P = W\) conjecture of de Cataldo-Hausel-Migliorini.

Abstract: 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.

Abstract: For any free oriented Borel–Moore homology theory \(A\), we construct an associative product on the \(A\)-theory of the stack of Higgs torsion sheaves over a projective curve \(C\). We show that the resulting algebra \(A\mathbf{Ha}_C^0\) admits a natural shuffle presentation, and prove it is faithful when \(A\) is replaced with usual Borel–Moore homology groups. We also introduce moduli spaces of stable triples, heavily inspired by Nakajima quiver varieties, whose \(A\)-theory admits an \(A\mathbf{Ha}_C^0\)-action. These triples can be interpreted as certain sheaves on \(\mathbb P(\omega_C\oplus \mathcal O_C)\). In particular, we obtain an action of \(A\mathbf{Ha}_C^0\) on the cohomology of Hilbert schemes of points on \(T^*C\).