Poisson Structures on the Conifold and Local Calabi-Yau Threefolds

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. 1. Motivation and results We are interested in holomorphic Poisson structures on Calabi–Yau threefolds that contain a contractible rational curve. Here we consider the local situation. Hence, we study Poisson structures on Calabi–Yau threefolds that are the total space of a rank 2 vector bundle on P1. A result of Jiménez [Jim] says that the contraction of a smooth rational curve on a threefold may happen in exactly 3 cases, namely when the normal bundle to such a curve is one of OP1(−1) ⊕ OP1(−1), OP1(−2) ⊕ OP1(0), or OP1(−3) ⊕ OP1(1), although only in the first case it can contract to an isolated singularity. In this work we describe completely the local case, that is, we classify all isomorphism classes of holomorphic Poisson structures on the local Calabi–Yau threefolds Wk := Tot(OP1(−k) ⊕ OP1(k − 2)), k = 1, 2, 3, calculate their Poisson cohomology, describe their symplectic foliations and some properties of their moduli. Polishchuk shows a correspondence between Poisson structures on a scheme X and a blow-up X̃, which applies to the cases we study [Po, Thm. 8.2, 8.4]. Hence, describing Poisson structures on Wk is equivalent to describing Poisson structures on the singular threefolds obtained from them by contracting the rational curve to a point. Poisson structures are parametrized by those elements of H(Wk, Λ2T Wk) which are integrable. We briefly recall some basic definitions from Poisson cohomology, for details see [LPV, Ch. 4]. Let (M, π) be a Poisson Manifold. The graded algebra X•(M) = Γ(∧•T M) and the degree-1 differential operator dπ = [π, ·] define the Poisson Cohomology of (M, π). The first cohomology groups have clear geometric meaning: H(M, π) = ker[π, ·] = Cas(π) = holomorphic functions on M which are constant along symplectic leaves. These are the Casimir functions of (M, π). H(M, π) = Poiss(π) Ham(π) is the quotient of Poisson vector fields by Hamiltonian vector fields. We compute Poisson cohomology groups and use them to distinguish Poisson structures, identifying their degeneracy loci. The r degeneracy locus of a holomorphic Poisson structure σ on a complex manifold or algebraic variety X is defined as D2r(σ) := {x ∈ X | rank σ(x) ≤ 2r} , where σ is viewed as a map T ∗ X → TX by contracting a 1-form with the bivector field σ. At a given point on a complex threefold a holomorphic Poisson structure has either rank 2, or rank 0. Therefore, for the threefolds Wk we name D(σ) := D0(σ) the degeneracy locus of σ, hence it consists of points where σ has rank 0.