Logarithmic Poisson Cohomology

• Logarithmic Poisson Cohomology is a framework for studying Poisson structures with logarithmic singularities along divisors, integrating log-symplectic forms and Lie–Rinehart algebras.
• It constructs a cochain complex with logarithmic differential forms and Chevalley–Eilenberg differentials, linking classical Poisson cohomology to its logarithmic counterpart via quasi-isomorphisms.
• The theory has key applications in prequantization and deformation analysis, with explicit computations showing how divisor topology influences the cohomological structure.

Logarithmic Poisson cohomology is the cohomological theory associated with Poisson structures that exhibit logarithmic singularities along divisors, generalizing classical Poisson cohomology to the context where the Poisson bivector degenerates in a controlled way. The structure of this cohomology is intimately linked to the geometry of log symplectic forms, Lie–Rinehart algebras, and the algebra of logarithmic differential forms, revealing rich interplay between Poisson geometry, singularity theory, and algebraic and differential topology (Dongho, 2010, Goto, 2015, Lanius, 2016).

1. Logarithmic Poisson Structures and the Logarithmic Complex

A logarithmic Poisson structure on a smooth variety $X$ (complex algebraic or analytic) with a normal-crossing divisor $D \subset X$ is a Poisson bivector $\pi$ that satisfies two requirements: (i) $\pi$ is a section of $\wedge^2 T_X(-\log D)$—vector fields tangent to $D$—and (ii) its Schouten–Nijenhuis bracket vanishes, $[\pi, \pi]=0$. The sheaf of logarithmic 1-forms $\Omega^1_X(\log D)$ consists locally of $\frac{dz_i}{z_i}$ for divisor variables and $dz_j$ for transverse variables, with dual sheaf $T_X(-\log D)$ for vector fields tangent to all branches of $D$ (Dongho, 2010).

The natural Lie–Rinehart algebra structure on $(\Omega^1_X(\log D), [\cdot, \cdot]_\pi)$ allows the definition of a logarithmic Hamiltonian map $H\colon \Omega^1_X(\log D) \to T_X(-\log D)$ by contraction with $\pi$. This representation underpins the construction of the logarithmic Poisson cochain complex

$$\left(\wedge^k_{\mathcal{O}_X} \Omega^1_X(\log D), \, d^{\log}_\pi\right)$$

with the differential $d^{\log}_\pi$ defined as a Chevalley–Eilenberg differential; its explicit formula, in terms of local generators, is recorded in (Dongho, 2010).

2. Cohomology Theory: Logarithmic Poisson Cohomology

The logarithmic Poisson cohomology $H^*_{\pi, \log}(X)$ is defined as the cohomology of the differential complex $(\wedge^*\Omega^1_X(\log D), d^{\log}_\pi)$. This structure generalizes classical Poisson cohomology, incorporating the sheaf-theoretic and geometric subtleties induced by the divisor $D$. The key features include:

• All Hamiltonian derivations in this setting are tangent to $D$.
• The complex is functorial with respect to the morphisms preserving the log structure.
• The cohomology groups can be computed locally using log-canonical coordinates, yielding computational tractability in many examples (Dongho, 2010).

A parallel, fully differentiable ($C^\infty$) framework exists, as developed in (Goto, 2015), for real manifolds and codimension-2 submanifolds $D \subset M$, with log-symplectic forms given locally by 2-forms with simple poles along $D$ and Poisson bivectors with corresponding degeneracy.

3. Comparison to Classical and Log-Symplectic Poisson Cohomology

A fundamental theorem establishes that when the underlying Poisson structure is log-symplectic—i.e., arises from a non-degenerate (up to simple poles) closed 2-form $\omega \in \Omega^2_X(\log D)$—there is a quasi-isomorphism between the classical Poisson cohomology complex and its logarithmic counterpart: $H^*_\pi(X) \simeq H^*_{\pi, \log}(X)$ This correspondence holds both in the holomorphic and $C^\infty$ (real) categories, with quasi-isomorphism realized via the identification $T_X(-\log D) \cong \Omega^1_X(\log D)$ supplied by the log-symplectic form (Dongho, 2010, Goto, 2015). The logarithmic Poisson cohomology thus recovers the de Rham cohomology of the complement $X \setminus D$ in many cases.

The existence of genuine differences is highlighted in examples where the Poisson structure is only logarithmic principal but not log-symplectic; in such cases, $H^*_\pi$ and $H^*_{\pi, \log}$ may differ, with $H^*_{\pi, \log}$ usually being “smaller” or recording only those classes that extend to the logarithmic structure (Dongho, 2010).

4. Computations and Examples

Detailed computations include:

Example Algebra/Manifold	Poisson Bracket	Log Structure	$H^*_\pi$	$H^*_{\pi, \log}$
$\mathbb{C}[x,y]$	$\{x,y\} = x$	$D = \{x=0\}$, log-sympl.	$1$, $1$, $0$	$1$, $1$, $0$
$\mathbb{C}[x,y]$	$\{x,y\} = x^2$	$D = \{x^2=0\}$, not log-s.	$1$, $1$–param., $0$	$1$, $1$–param., $0$
$\mathbb{C}[x,y,z]$	$\{y,z\}=x\,y\,z$	$D=\{xyz=0\}$, principal	$\dim$ large	strictly smaller

On complex surfaces with smooth anticanonical divisors, e.g., del Pezzo or Hirzebruch surfaces, the Poisson cohomology coincides with the de Rham cohomology of the complement, with explicit group ranks dependent on the topology of $S \setminus D$ (Goto, 2015). On partitionable log-symplectic manifolds, the Poisson cohomology splits as a direct sum involving the $b$-de Rham cohomology and summands from all possible intersections of divisor strata, as proved in (Lanius, 2016).

5. Modification of Logarithmic Differential Forms and Saito’s Hypothesis

The definition of logarithmic forms is subtle: the original notion of K. Saito allowed for non-reduced divisors, but this can yield pathological bases for the module of logarithmic forms, e.g., incorrectly including $dy$ along $x^2=0$. Therefore, it is crucial to require the defining equation of the divisor $D$ be reduced for the correct logarithmic theory in Poisson geometry (Dongho, 2010). This restriction ensures the log vector fields and the corresponding log Poisson cohomology reflect the true geometry of the divisor and its singularities.

6. Structural Decomposition and Rigged de Rham Complexes

For manifolds with normal crossing divisors $D = \{Z_1,\ldots, Z_r\}$, the sheaf of log-differential forms $\Omega^*(M, \log D)$ and the log-tangent bundle ${}^bTM$ induce a nested filtration of Lie algebroids, each step corresponding to rescaling along a divisor component. Lanius constructs a direct-sum decomposition in the Poisson cohomology,

$$H^p_\pi(M) \cong {}^b H^p(M) \oplus \bigoplus_{\text{divisor intersections}} H^{p-m}(V_{I,J,K,L}; \dots)$$

where $V_{I,J,K,L}$ are intersection strata and the summands record contributions from local residue structures and normal crossings (Lanius, 2016). This explicit decomposition highlights the connection between the cohomology and the combinatorics of the stratification induced by $D$.

7. Applications and Prequantization

Logarithmic Poisson cohomology is central to prequantization problems on singular Poisson spaces. For instance, the logarithmic Poisson algebra $(\mathbb{C}[x,y], \{x,y\}=x)$ admits a central extension whose curvature class equals the Poisson bivector. If $H^2_{\pi, \log}=0$, as in this example, there exists a projective module with a log-connection (prequantum line bundle), realizing a representation of the log–Lie–Rinehart algebra (Dongho, 2010). More broadly, these cohomological structures control deformations of generalized complex structures induced by log-symplectic forms. For complex surfaces with smooth anti-canonical divisors, all such deformations are unobstructed and parametrized by the second de Rham cohomology of the complement (Goto, 2015).

The framework highlights the necessity of the divisor's smoothness: introducing singularities into $D$ yields extra cohomological contributions, often corresponding to the Tjurina algebra of singular points, hence encoding subtle information about both the geometry of the degeneracy locus and the global topology (Goto, 2015). These results directly inform the structure and classification of singular Poisson and generalized complex manifolds.

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References:

• Dongho, "Logarithmic Poisson cohomology: example of calculation and application to prequantization" (Dongho, 2010)
• Goto, "Unobstructed deformations of generalized complex structures induced by $C^\infty$ logarithmic symplectic structures and logarithmic Poisson structures" (Goto, 2015)
• Lanius, "Poisson cohomology of a class of log symplectic manifolds" (Lanius, 2016)