Exact formulas for invariants of Hilbert schemes

Abstract

A theorem of Göttsche establishes a connection between cohomological invariants of a complex projective surface S and corresponding invariants of the Hilbert scheme of n points on S. This relationship is encoded in certain infinite product q-series which are essentially modular forms. Here we make use of the circle method to arrive at exact formulas for certain specializations of these q-series, yielding convergent series for the signature and Euler characteristic of these Hilbert schemes. We also analyze the asymptotic and distributional properties of the q-series' coefficients.

From partitions to Hodge numbers of Hilbert schemes of surfaces

“…In these cases we will use the circle method to obtain exact formulae, as well as asymptotic and distributional information, for these Hodge numbers for a certain class of complex projective surfaces. This work generalizes previous work by some of the authors [14].…”

“…is a Kloosterman sum. Integrating as in [6], we find that for some δ > 0, where we define the scaled modified Bessel function of the first kind (3.7) I * (ι 1 , ι 2 , j, k; n) := 2πn − πχ(S) 12…”

“…Remark. The proof of Theorem 5 differs most from the proof of the corresponding statement in [6] in the establishment of the Kloosterman sum bound…”

“…By specializing (1.7) appropriately, we obtain generating functions for a variety of topological invariants (see [6]). In order to study the distributional properties of Hodge numbers, we consider…”

“…Building on work in [12], from which it follows that for S a K3 surface γ S (r, ℓ, 0, 1; n) ∼ γ S (r ′ , ℓ, 0, 1; n) as n −→ ∞, and [6], which described equidistribution in the case ℓ 1 = ℓ 2 = 2, we use the asymptotics derived from the exact formula in Theorem 5 to make the following statement about the equidistribution of Hodge numbers for appropriate surfaces: Theorem 6. Let S be a smooth projective complex surface such that χ(S) ≥ σ(S).…”