[Paper Review] Hilbert schemes, Hecke algebras and the Calogero-Sutherland system

Kevin Costello, I. Grojnowski|ArXiv.org|Oct 13, 2003

Algebraic structures and combinatorial models19 references19 citations

TL;DR

This paper establishes a novel link between the cohomology ring of Hilbert schemes of points on a smooth algebraic surface and integrable systems by identifying the ring structure as the centralizer of a deformed Calogero-Sutherland Hamiltonian in a ring of differential operators. Using a Fock space model and Dunkl-Cherednik operators, it constructs an explicit commuting family of differential operators that realize cup product multiplication, providing a complete algebraic characterization of the Hilbert scheme cohomology ring via integrals of motion.

ABSTRACT

We describe the ring structure of the cohomology of the Hilbert scheme of points for a smooth surface X. When the canonical class K_X = 0, this was done by Lehn and Sorger, extending earlier work when X = C^2. Their approach does not generalize. Instead we recast this problem as a question of finding integrals of motion for a Hamiltonian which describes ``intersection with the boundary''. To do so, we use the identification of the cohomology of the Hilbert schemes with a Fock space modelled on the lattice H(X). With this identification, Lehn computed the operator of intersection with the boundary. It is essentially the Calogero-Sutherland Hamiltonian. We then solve the problem of finding integrals of motion by using the Dunkl-Cherednik operators to find an explicit commuting family of differential operators; these operators represent cup product on the Hilbert scheme. We provide two characterizations of the Hilbert scheme multiplication operators; the first as an algebra of operators that can be inductively built from functions and the CS Hamiltonian, and the second in terms of the centralizer of the CS Hamiltonian inside an appropriate ring of differential operators.

Motivation & Objective

• To solve the long-standing problem of describing the ring structure on the cohomology of Hilbert schemes of points on a smooth algebraic surface, especially when the canonical class is non-zero.
• To overcome the failure of previous approaches—based on orbifold cohomology and Hecke algebras—that only work when the canonical class vanishes.
• To reframe the cohomology ring structure as a problem in integrable systems by identifying the boundary intersection operator as a variant of the Calogero-Sutherland Hamiltonian.
• To construct an explicit, algebraic realization of the multiplication operators on the Hilbert scheme cohomology using differential operators commuting with this Hamiltonian.
• To provide a presentation of the cohomology ring independent of the existence of the Hilbert scheme, using only the cohomology of the surface and its canonical class.

Proposed method

• Realize the cohomology of the Hilbert scheme as a Fock space isomorphic to the symmetric algebra of the cohomology of the surface, using the Heisenberg algebra action.
• Identify the operator of cup product with the boundary divisor as a second-order differential operator equivalent to a deformed Calogero-Sutherland Hamiltonian.
• Use the Dunkl-Cherednik operators to construct a commuting family of differential operators that serve as integrals of motion for this Hamiltonian.
• Define a ring of continuous differential operators on the Fock space depending on the Frobenius algebra structure of the surface's cohomology and a parameter related to the canonical class.
• Characterize the algebra of multiplication operators as the centralizer of the Calogero-Sutherland Hamiltonian in the ring of differential operators, and show it equals the algebra generated by the integrals of motion.
• Prove that the associated graded algebra of the multiplication operator algebra is isomorphic to the symmetric algebra on the cohomology and momentum variables, confirming its completeness.

Experimental results

Research questions

• RQ1 How can the cohomology ring structure of the Hilbert scheme of points on a smooth surface be described algebraically when the canonical class is non-zero?
• RQ2 Can the cup product multiplication on the cohomology of the Hilbert scheme be realized as a commuting family of differential operators?
• RQ3 Is there a way to construct the cohomology ring of the Hilbert scheme without relying on the geometric existence of the Hilbert scheme itself?
• RQ4 Can the problem of determining the cohomology ring be reformulated as a problem in integrable systems, specifically via the Calogero-Sutherland model?
• RQ5 What is the precise algebraic structure of the centralizer of the boundary intersection operator in the ring of differential operators on the Fock space?

Key findings

• The algebra of multiplication operators on the cohomology of the Hilbert scheme is isomorphic to the centralizer of the Calogero-Sutherland Hamiltonian in the ring of differential operators on the Fock space.
• The constructed algebra of differential operators, denoted $\mathcal{D}^{\text{hilb}}$, coincides with the algebra of left multiplication operators on each $\mathcal{F}^n$.
• The associated graded algebra of $\mathcal{D}^{\text{hilb}}$ is isomorphic to $\operatorname{Sym}H[p]$, the symmetric algebra on the cohomology and momentum variables, confirming completeness.
• The integrals of motion for the Calogero-Sutherland Hamiltonian are explicitly constructed via Dunkl-Cherednik operators and form a commuting family that generates the multiplication algebra.
• The construction provides a presentation of the cohomology ring independent of the Hilbert scheme, depending only on the weak Frobenius algebra structure of $H(X,\mathbb{C})$ and the canonical class $K\in H^2(X,\mathbb{C})$.
• The method resolves the obstruction in previous approaches when $K \neq 0$, by replacing orbifold cohomology with an integrable systems framework.