Poisson structures of divisor-type

Abstract

Poisson structures of divisor-type are those whose degeneracy can be captured by a divisor ideal, which is a locally principal ideal sheaf with nowhere-dense quotient support. This is a large class of Poisson structures which includes all generically-nondegenerate Poisson structures, such as log-, $b^k$-, elliptic, elliptic-log, and scattering Poisson structures. Divisor ideals are used to define almost-injective Lie algebroids of derivations preserving them, to which these Poisson structures can be lifted, often nondegenerately so. The resulting symplectic Lie algebroids can be studied using tools from symplectic geometry. In this paper we develop an effective framework for the study of Poisson structures of divisor-type (also called almost-regular Poisson structures) and their Lie algebroids. We provide lifting criteria for such Poisson structures, develop the language of divisors on smooth manifolds, and discuss residue maps to extract information from Lie algebroid forms along their degeneraci loci.

arXiv comment57 pages

References in corpus (9)

• Symplectic and Poisson geometry on b-manifolds
• Poisson modules and degeneracy loci
• Stable generalized complex structures
• Type one generalized Calabi--Yaus
• Geometric structures and Lie algebroids
• On Integrable Systems & Rigidity for PDEs with Symmetry
• Modular Classes of Lie Groupoid Representations up to Homotopy
• Poisson cohomology of a class of log symplectic manifolds
• Obstructions for Symplectic Lie Algebroids

Cited by in corpus (2)

• Linearization of Poisson groupoids
• Obstructions for Symplectic Lie Algebroids