On the Motive of the Hilbert scheme of points on a surface

Abstract

The Hilbert scheme S[n] of points on an algebraic surface S is a simple example of a moduli space and also a nice (crepant) resolution of singularities of the symmetric power S (n) . For many phenomena expected for moduli spaces and nice resolutions of singular varieties it is a model case. Hilbert schemes of points have connections to several fields of mathematics, including moduli spaces of sheaves, Donaldson invariants, enumerative geometry of curves, infinite dimensional Lie algebras and vertex algebras and also to theoretical physics. This talk will try to give an overview over these connections. IntroductionThe Hilbert scheme S[n] of points on a complex projective algebraic surface S is a a parameter variety for finite subschemes of length n on S. It is a nice (crepant) resolution of singularities of the n-fold symmetric power S (n) of S. If S is a K3 surface or an abelian surface, then S[n] is a compact, holomorphic symplectic (thus hyperkähler) manifold. Thus S [n] is at the same time a basic example of a moduli space and an example of a nice resolution of singularities of a singular variety. There are a number of conjectures and general phenomena, many of which originating from theoretical physics, both about moduli spaces for objects on surfaces and about nice resolutions of singularities. In all of these the Hilbert scheme of points can be viewed as a model case and sometimes as the main motivating example. Hilbert schemes of points on a surface have connections to many topics in mathematics, including moduli spaces of sheaves and vector bundles, Donaldson invariants, Gromov-Witten invariants and enumerative geometry of curves, infinite dimensional Lie algebras and vertex algebras, noncommutative geometry and also theoretical physics.

Vertex algebras and the cohomology ring structure of Hilbert schemes of points on surfaces

“…In particular, Lehn constructed the Virasoro algebra in a geometric fashion and studied certain tautological sheaves over X [n] . Some other recent work on Hilbert schemes includes [dCM,EGL,Go2,Hai,LQZ,Wan].…”

Counting polynomials over finite fields with given root multiplicities

“…As [Sym r A d ] = [A rd ] (e.g. see [Göt01,Lemma 4.4]), Theorem 4.2 (2) with X = A d gives a very similar result to that of X = A 1 , just with each [A s ] replaced by [A sd ], and we have for any d ≥ 0. These are the cases that motivate Conjecture 1.2 (see [VW12, 1.41-1.44] for more details about this motivation).…”

Counting Real Rational Curves onK3 Surfaces

“…The structure of the latter one is exactly the same as in the motivic formula for the Hilbert schemes of points of complex surfaces (see, e.g., [9] or [15]), so the standard proofs apply and give us the formula…”