[Paper Review] Poisson cohomology of a class of log symplectic manifolds

Melinda Lanius|arXiv (Cornell University)|May 12, 2016

Geometric and Algebraic Topology8 references4 citations

TL;DR

This paper computes the Poisson cohomology of a class of log symplectic manifolds with transverse hypersurfaces, using a rigged algebroid approach to generalize b-symplectic cohomology. The key result identifies Poisson cohomology as a direct sum over intersections of divisors, incorporating twisted cohomology terms with canonical line bundles, extending known results for b-symplectic structures.

ABSTRACT

We compute the Poisson cohomology of a class of Poisson manifolds that are symplectic away from a collection $D$ of hypersurfaces. These Poisson structures induce a generalization of symplectic and cosymplectic structures, which we call a k-cosymplectic structure, on the intersection of hypersurfaces in $D$.

Motivation & Objective

• To compute the Poisson cohomology of log symplectic manifolds with a normal crossing divisor D, extending known results for b-symplectic structures.
• To generalize the cosymplectic structure on singular hypersurfaces to higher codimension intersections in the log symplectic setting.
• To develop a cohomological framework using the rigged algebroid to handle the singular Poisson structures induced by log symplectic forms.
• To identify the precise decomposition of Poisson cohomology in terms of de Rham cohomology of the ambient manifold and twisted cohomology on divisor intersections.

Proposed method

• Uses the rigged algebroid construction to model Poisson cohomology via Lie algebroid cohomology of the b-tangent bundle over a manifold with a normal crossing divisor D.
• Applies the Lie algebroid cohomology isomorphism from [6] to express b-cohomology as a direct sum over intersections of divisor components.
• Introduces a filtration on the complex of log forms with respect to defining functions of the divisor components, enabling inductive computation of cohomology.
• Employs a change-of-variables argument for defining functions of the divisor components to show invariance of the cohomology classes under coordinate transformations.
• Derives the cohomology of the log symplectic complex by analyzing the kernel and image of the differential in a local model near divisor intersections.
• Establishes that the Poisson cohomology splits as a sum over all possible intersections of divisor components, with terms involving twisted cohomology on each stratum.

Experimental results

Research questions

• RQ1 How does the Poisson cohomology of a log symplectic manifold with a normal crossing divisor D decompose in terms of the cohomology of the ambient manifold and its strata?
• RQ2 What generalization of cosymplectic structures arises on the transverse intersections of the divisor components in the log symplectic setting?
• RQ3 Can the Poisson cohomology be computed via a Lie algebroid structure on the b-tangent bundle, and how does this compare to the b-symplectic case?
• RQ4 What role do canonical line bundles over divisor intersections play in the cohomological decomposition of log symplectic Poisson structures?
• RQ5 How do changes in defining functions of the divisor components affect the cohomology classes in the Poisson complex?

Key findings

• The Poisson cohomology of the log symplectic manifold is isomorphic to the direct sum of the de Rham cohomology of the ambient manifold and twisted cohomology groups on all strata of the divisor intersection lattice.
• For each stratum defined by the intersection of divisor components, the cohomology includes terms involving the dual of the conormal bundle of the stratum, specifically |N^*Z|^{-1} for each component.
• The cohomology decomposition includes contributions from all possible subsets of divisor components, with the total degree shift depending on the number and type of components involved in each intersection.
• The cohomology on each stratum is invariant under change of defining functions for the divisor components, ensuring a well-defined geometric structure.
• The final result generalizes the b-symplectic cohomology formula H^p_π(M) ≃ H^p(M) ⊕ H^{p-1}(Z) to a full decomposition over all divisor intersections with twisted coefficients.
• The cohomology is fully characterized by the formula: ^bH^p(M) ≃ H^p(M) ⊕ ⨁_{I,J,K,L} H^{p-m}(∩Z; ⊗|N^*Z|^{-1}) where m = 2|I| + |J| + |K| + |L| and I,J,K,L are index sets satisfying I ≠ ∅ and I ∩ J = I ∩ K = ∅.