Elliptic log symplectic brackets on projective bundles

TL;DR

The paper develops a systematic method to construct elliptic log symplectic Poisson brackets on projective bundles by deforming semi-toric log symplectic forms on $\mathsf{X}=\mathbb{P}^{n-1} \times \mathbb{D}^m$ through smoothing diagrams. It hinges on the combinatorics of smoothable edges and proves that deformations associated to cycles of these edges are jointly unobstructed, enabling explicit elliptic brackets such as the Feigin-Odesskii family $q_{n,k}$ on $\mathbb{P}^{n-1}$. By analyzing concrete examples, including $\mathcal{C}_{4,1}$ and $\mathcal{X}_5$, the work demonstrates that elliptic log symplectic structures arise naturally from the smoothing-cycle framework and provides evidence for a broader conjecture that all elliptic log symplectic brackets on $\mathbb{P}^{n-1}\times \mathbb{D}^m$ are captured by this method. The results bridge deformation theory, combinatorial biresidue data, and elliptic Poisson geometry with potential connections to elliptic $R$-matrices and quantization.

Abstract

Let $\mathsf{X}$ be the product of a complex projective space and a polydisc. We study Poisson brackets on $\mathsf{X}$ that are log symplectic, that is, generically symplectic and such that the inverse two-form has only first order poles. We propose a method of constructing such Poisson brackets that additionally are elliptic, in a precise sense. Our method relies on the local Torelli theorem for log symplectic manifolds of Pym, Schedler and the author, and uses combinatorics of smoothing diagrams. We demonstrate effectiveness of the method on a series of examples, recovering, in particular, all log symplectic cases of elliptic Feigin-Odesskii Poisson brackets $q_{n,k}$ on $\mathbb{P}^{n-1}$.

Figures (6)

• Figure 1: Two types of colored angles
• Figure 2: Effect of the Poisson deformation in \ref{['prop:smoothable_edge_deform']} on the polar divisor near a generic point of $\{y_i=y_j=0\}$.
• Figure 3: Degeneracy divisor of the Poisson deformation $\pi(\varepsilon)$
• Figure 4: Examples of smoothing diagrams having full smoothable cycles. The vertices are labeled counterclockwise, the bottom horizontal edge being $(n-1) \,\, 0$.
• Figure 5: Degeneracy divisor for the Poisson deformation $\mathcal{C}_{4,1}$.

Theorems & Definitions (35)

• Definition A
• Conjecture A
• Theorem A
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Example 1