Poisson structures on the conifold and local Calabi-Yau threefolds

TL;DR

This work classifies holomorphic Poisson structures on local Calabi–Yau threefolds $W_k$ ($k=1,2,3$) presented as total spaces $\mathrm{Tot}(\mathcal{O}_{\mathbb{P}^1}(-k) \oplus \mathcal{O}_{\mathbb{P}^1}(k-2))$, computes their Poisson cohomology, and analyzes the resulting symplectic foliations and moduli. The authors construct explicit generators for $H^0(W_k, \Lambda^2 TW_k)$ (four for $W_1$, five for $W_2$, and thirteen for $W_3$) and determine integrability conditions; they show symmetries and principal embeddings of Poisson surfaces generate all Poisson structures in each case. They distinguish isomorphism classes via degeneracy loci and Casimir functions, and they provide a detailed description of the symplectic foliations, including explicit embeddings of Poisson surfaces that generate the full Poisson structure space. The results illuminate how holomorphic Poisson geometry encodes noncommutative deformations and moduli for these local CY threefolds, with applications to contraction phenomena and geometric transitions in the conifold context.

Abstract

We describe bivector fields and Poisson structures on local Calabi-Yau threefolds which are total spaces of vector bundles on a contractible rational curve. In particular, we calculate all possible holomorphic Poisson structures on the conifold.

Theorems & Definitions (62)

• Remark 1.1
• Proposition 1.2
• proof
• Lemma : \ref{['biW1']}
• Theorem : \ref{['t1']}
• Theorem : \ref{['iso']}
• Theorem : \ref{['w1fol']}
• Theorem : \ref{['princ']}
• Lemma : \ref{['W2gens']}
• Theorem : \ref{['foliations2']}