[Paper Review] Deformations of holomorphic pseudo-symplectic Poisson manifolds

Ziv Ran|arXiv (Cornell University)|Aug 12, 2013

Geometry and complex manifolds11 references3 citations

TL;DR

This paper establishes unobstructed deformations for weakly P-normal holomorphic Poisson manifolds—compact Kähler manifolds with a Poisson structure degenerating along a divisor with mild singularities. Using differential graded Lie algebras and mixed Hodge theory, it proves that all Poisson deformations are unobstructed, extending unobstructedness results to Hilbert schemes of del Pezzo surfaces and other examples with logarithmic symplectic structures.

ABSTRACT

We prove unobstructed deformations for compact Kaehlerian even-dimensional Poisson manifolds whose Poisson tensor degenerates along a divisor with mild singularities. Examples include Hilbert schemes of del Pezzo surfaces.

Motivation & Objective

• To establish unobstructed deformation theory for holomorphic Poisson manifolds whose Poisson tensor degenerates along a divisor with mild singularities.
• To generalize unobstructedness results beyond symplectic and Fano cases to a broad class of pseudo-symplectic Poisson structures.
• To provide a deformation-theoretic framework for Hilbert schemes of del Pezzo surfaces equipped with induced Poisson structures.
• To clarify the relationship between Poisson deformations and locally trivial deformations of the degeneracy divisor.
• To introduce and study the notion of weak P-normality as a natural generalization of P-normality for Poisson structures with controlled singularities.

Proposed method

• Introduces the concept of weak P-normality via codimension conditions on degeneracy loci Pf^k(Π) and transversality of branches.
• Applies differential graded Lie algebra (dgla) techniques to the tangent sheaf with logarithmic singularities, T_X⟨log D⟩.
• Uses Deligne’s E1-degeneration and the Cartan formula to show vanishing of the obstruction pairing in H^2(Ω^{n-1}_X⟨log D⟩).
• Relies on the isomorphism T_X ≅ Ω^{n-1}_X(D) for Kähler manifolds with anticanonical divisor D to relate Poisson and logarithmic geometry.
• Employs iterated blowups along the flag of degeneracy loci to analyze the structure of the Pfaffian divisor and its total transform.
• Computes explicit local normal forms for P-normal Poisson structures and verifies their compatibility with weak P-normality.

Experimental results

Research questions

• RQ1 Under what conditions on the degeneracy divisor D of a holomorphic Poisson structure is the full Poisson deformation space unobstructed?
• RQ2 Can unobstructedness of Poisson deformations be established for manifolds where D is reduced with normal crossings but not smooth?
• RQ3 To what extent do locally trivial deformations of the pair (X,D) lift to Poisson deformations of (X,Π,D)?
• RQ4 What is the precise geometric condition on the degeneracy locus that ensures unobstructedness in the Poisson deformation space?
• RQ5 Do Hilbert schemes of del Pezzo surfaces with induced Poisson structures satisfy unobstructed Poisson deformation theory?

Key findings

• The full Poisson deformation space Def(X,Π) is smooth for any weakly P-normal Poisson manifold (X,Π), proving unobstructed deformations.
• Poisson deformations of (X,Π) induce locally trivial deformations on the Pfaffian divisor D, and every such locally trivial deformation of (X,D) lifts to a deformation of (X,Π,D).
• The forgetful morphisms from the deformation space of the triple (X,Π,D) to Def(X,Π) and to Def_loc.trivial(X,D) are both smooth.
• The result applies to Hilbert schemes S^{[r]} of del Pezzo surfaces when the anticanonical curve is smooth, yielding unobstructed Poisson deformations.
• P-normal Poisson structures admit a local normal form involving x_i ∂/∂x_i ∧ ∂/∂y_i and standard symplectic terms, with corank 2 precisely at smooth points of D.
• The paper shows that P-normality implies weak P-normality, and provides a framework for extending unobstructedness results to broader classes of Poisson manifolds with singular Pfaffian divisors.