The asymptotic profile of $\chi_y$-genera of Hilbert schemes of points on K3 surfaces

Abstract

Abstract. The Hodge numbers of the Hilbert schemes of points on algebraic surfaces are given by Göttsche's formula, which expresses the generating functions of the Hodge numbers in terms of theta and eta functions. We specialize in this paper to generating functions of the χ y -genera of Hilbert schemes of n points on K3 surfaces. We determine asymptotic values of the coefficients of the χ y -genus for n → ∞ as well as their asymptotic profile.

Exact formulas for invariants of Hilbert schemes

“…Note that when S is a K3 surface, Corollary 1.3 (1) recovers the equidistribution of the b * S (r; n) that follows from the work of Manschot and Zapata Rolon in [14].…”

“…Manschot and Zapata Rolon [14] found for K3 surfaces S that if m is fixed, then as n → ∞ we have b S (m; n) ∼ π 3 √ 2 n − 29 4 · exp(4π √ n).…”

“…Göttsche stated a more refined infinite product formula concerning Hodge numbers, which are cohomological invariants that can be assembled into Betti numbers and the Euler characteristic. Recently, Manschot and Zapata Rolon [14] studied the asymptotic distribution of linear combinations of these Hodge numbers, which are realized as coefficients of Laurent polynomials called the χ y genera for K3 surfaces (see (2.4) for a definition of χ y ). The χ y genera can also be assembled using the generating function Akin to (1.1), Göttsche also provides an infinite product for Y S (y; τ ) (see Lemma 2.2).…”

From partitions to Hodge numbers of Hilbert schemes of surfaces

“…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).…”

Elliptic Genera and 3d Gravity

“…See [27] for more information on the order O(log N ) corrections. For terms with polarity close to zero, the O(log N ) corrections are less important, and we see that the bound is subsaturated as expected.…”