[Paper Review] Tau function and the Prym class

D. Korotkin, P. Zograf|arXiv (Cornell University)|Feb 4, 2013

Meromorphic and Entire Functions4 citations

TL;DR

This paper introduces two Bergman tau functions on the moduli space of holomorphic quadratic differentials and uses their asymptotic behavior near boundary divisors to derive explicit relations in the rational Picard group. The key result is a new relation between the Prym class, Hodge class, tautological class, and boundary divisors, which recovers and extends Mumford's relation in the context of quadratic differentials via tau function analysis.

ABSTRACT

We use the formalism of the Bergman tau functions to study the geometry of moduli spaces of holomorphic quadratic differentials on complex algebraic curves. We introduce two natural tau functions and interpret them as holomorphic sections of certain line bundles on the moduli space. Analyzing the asymptotic behavior of these tau functions near the boundary of the moduli space we get two non-trivial relation in the rational Picard group of the moduli space of quadratic differential.

Motivation & Objective

• To study the geometry of the moduli space $\overline{\mathcal{Q}}_g$ of holomorphic quadratic differentials on stable curves of genus $g \geq 2$.
• To define and analyze two natural tau functions as holomorphic sections of line bundles on $P\overline{\mathcal{Q}}_g$.
• To derive relations in the rational Picard group of $P\overline{\mathcal{Q}}_g$ using asymptotic analysis of these tau functions near boundary divisors.
• To establish a new relation between the Prym class $\lambda_P$, the Hodge class $\lambda$, the tautological class $\psi$, and boundary divisor classes $\delta_j$.

Proposed method

• The authors define two tau functions $\tau_\pm$ using the Bergman tau function formalism on the moduli space of quadratic differentials.
• They analyze the asymptotic behavior of $\tau_\pm$ near the boundary divisors $D_{\text{deg}}, D_0, \dots, D_{[g/2]}$ of $P\overline{\mathcal{Q}}_g$ using local coordinates and residue computations.
• The analysis involves lifting quadratic differentials to abelian covers $\widehat{C} \to C$ via the canonical double cover associated with $q = v^2$, and decomposing the Hodge bundle on $\widehat{C}$ into $\mu$-invariant and anti-invariant parts.
• The Prym bundle $\Lambda_-$ is identified as the $-1$ eigenspace of the involution $\mu$, and its first Chern class $\lambda_P$ is computed via the tau function asymptotics.
• The method relies on residue calculations of the difference of Schwarzian derivatives $S_{\widehat{B}_0} - S_{v_0}$ near nodes and poles.
• The resulting asymptotics are used to derive linear relations in the rational Picard group by matching logarithmic derivatives of $\tau_\pm$ to divisor classes.

Experimental results

Research questions

• RQ1 How do the Bergman tau functions behave asymptotically near the boundary of the moduli space $P\overline{\mathcal{Q}}_g$ of quadratic differentials?
• RQ2 What relations in the rational Picard group of $P\overline{\mathcal{Q}}_g$ can be derived from the asymptotic behavior of these tau functions?
• RQ3 How is the Prym class $\lambda_P$ related to the Hodge class $\lambda$, the tautological class $\psi$, and the boundary divisor classes $\delta_j$?
• RQ4 Can the Mumford relation $\lambda_2 = 13\lambda_1$ be recovered and extended in the context of the moduli space of quadratic differentials?

Key findings

• The asymptotic behavior of the tau functions $\tau_\pm$ near the degenerate divisor $D_{\text{deg}}$ yields the relation $48\lambda - \frac{20}{3}(g-1)\psi = \frac{2}{3}\delta_{\text{deg}} + 4\sum_{j=0}^{[g/2]}\delta_j$.
• Near the divisor $D_0$, the asymptotics of $\tau_\pm$ lead to the same relation, confirming consistency across boundary components.
• Near each $D_j$ for $j \geq 1$, the asymptotics show $\tau_\pm \sim t^4$ as $t \to 0$, leading to the same global relation.
• The combination of all asymptotic results yields the key relation $\lambda_P - 13\lambda = -\sum_{j=0}^{[g/2]}\delta_j - \frac{3}{2}(g-1)\psi$ in $\text{Pic}(P\overline{\mathcal{Q}}_g) \otimes \mathbb{Q}$.
• This relation implies the pullback of Mumford's relation: $p^*\lambda_2 - 13p^*\lambda_1 = -\sum_{j=0}^{[g/2]}\delta_j$ on $P\overline{\mathcal{Q}}_g$, where $\lambda_2 = c_1(\pi_*\omega_g^2)$.
• The Prym class $\lambda_P$ is shown to satisfy $\lambda_P = p^*\lambda_2 - \frac{3g-3}{2}\psi$, linking it to the determinant bundle of quadratic differentials.