[Paper Review] Motivic classes of generalized Kummer schemes via relative power structures

Andrew Morrison, Junliang Shen|arXiv (Cornell University)|May 12, 2015

Nonlinear Waves and Solitons15 references3 citations

TL;DR

This paper develops a relative power structure over the Grothendieck ring of varieties relative to an abelian monoid to compute motivic classes of generalized Kummer schemes. It generalizes Cheah’s formula for Hilbert schemes and proves Gulbrandsen’s conjecture on Euler characteristics, while providing explicit formulas for virtual motives of Kummer schemes in surfaces and threefolds, including a new formula for the $χ_y$ genus in the 3-fold case with $y\leftrightarrow y^{-1}$ symmetry.

ABSTRACT

We develop a power structure over the Grothendieck ring of varieties relative to an abelian monoid, which allows us to compute the motivic class of the generalized Kummer scheme. We obtain a generalized version of Cheah's formula for the Hilbert scheme of points, which specializes to Gulbrandsen's conjecture for Euler characteristics. Moreover, in the surface case we prove a conjecture of Göttsche for geometrically ruled surfaces, and we obtain an explicit formula for the virtual motive of the generalized Kummer scheme in dimension three.

Motivation & Objective

• To develop a relative power structure over the Grothendieck ring of varieties over an abelian monoid to study generalized Kummer schemes.
• To generalize Cheah’s formula for Hilbert schemes of points to the motivic setting.
• To prove Gulbrandsen’s conjecture on the Euler characteristic of generalized Kummer schemes.
• To compute the virtual motive of generalized Kummer schemes in dimension three.
• To extend results to motivic classes of stacks of torsion sheaves on curves, surfaces, and threefolds.

Proposed method

• Introduce a relative power structure on the Grothendieck ring $K_0(\text{Var}/A)$ for an abelian variety $A$.
• Use the Hilbert scheme of $n$ points on a fibration $X\to A$ and its Hilbert–Chow map to define the generalized Kummer scheme as the fiber over zero.
• Apply Lemma 2.8 to pull back the motivic class of the Hilbert scheme over $A$ to obtain the motivic class of $K_n(X)$.
• Leverage virtual motives in $K_0(\text{HS})[\mathbb{L}^{-1/2}]$ for locally complete intersection schemes, particularly in dimension three.
• Use generating series $\mathcal{T}_X(t)$ and $\mathcal{H}_g^{\text{vir}}(t)$ to encode motivic classes of torsion sheaf stacks.
• Apply symmetric power structures and Young subgroup invariants to extract the Kummer fiber via $0^*$-pullback and $g(\alpha)$-weighted terms.

Experimental results

Research questions

• RQ1 Can a relative power structure over the Grothendieck ring of varieties over an abelian monoid be developed to compute motivic classes of generalized Kummer schemes?
• RQ2 Does the generalized Kummer scheme’s motivic class admit a formula analogous to Cheah’s for Hilbert schemes?
• RQ3 Can Gulbrandsen’s conjecture on the Euler characteristic of $K_n(X)$ be proven via motivic methods?
• RQ4 What is the explicit formula for the virtual motive of the generalized Kummer scheme in dimension three?
• RQ5 How do the Hodge–Deligne polynomials and $\chi_y$ genera of $K_n(X)$ reflect symmetries such as $y \leftrightarrow y^{-1}$?

Key findings

• The paper proves a generalized version of Cheah’s formula for the motivic class of the Hilbert scheme of points, now in the relative setting over an abelian variety.
• It establishes a new formula for the virtual motive of the generalized Kummer scheme in dimension three, extending previous results to non-smooth cases.
• For surfaces, the formula recovers Göttsche’s conjecture and confirms Göttsche–Soergel’s result for $g=2$, while proving Göttsche’s conjecture for $g=1$.
• In the 3-fold case with $X = K3 \times E$, the $\chi_y$ genus of $[K_n(X)]_{\text{vir}}$ is shown to satisfy $y \leftrightarrow y^{-1}$ symmetry: $\chi_{-y}([K_n(X)]_{\text{vir}}) = (y^{-n/2} + \cdots + y^{n/2}) \sum_{d\cdot m=n} d^2(y^{-m/2} + 22 + y^{m/2})$.
• The virtual motive of $K_n(X)$ for $X$ a 3-fold is given by $[\mathcal{T}_n(X)\to A]_{\text{vir}} = \sum_{\sum m r_i^m = n} \prod_{m\geq 1} m_*\left( \prod_{i=1}^{l_m} \mathbb{L}^{-r_i^m} \frac{[\text{Sym}^{r_i^m}(X)\to A]}{\mathbb{L}^{s_i(r_{\bullet}^m)} - 1} \right)$.