Four-dimensional conical symplectic hypersurfaces

Abstract

We show that every indecomposable conical symplectic hypersurface of dimension four is isomorphic to the known one, namely, the Slodowy slice X n which is transversal to the nilpotent orbit of Jordan type [2n − 2, 1, 1] in the nilpotent cone of sp 2n for some n ≥ 2. In the appendix written by Yoshinori Namikawa, conical symplectic varieties of dimension two are classified by using contact Fano orbifolds. Conical symplectic varieties as graded Poisson algebrasAn affine symplectic varietya finitely generated graded C-algebra with R 0 = C and the algebraic symplectic form ω on the regular part X reg is homogeneous with respect to the C * -action λ : C * × X → X coming from the grading on R, that is, there exists an integer s such that λ(t) * (ω) = t s ω for any t ∈ C * . By [LNSvS, Lemma 2.2], the weight s is positive.Let X = Spec R be a conical symplectic variety. The symplectic form ω is regarded as an isomorphism ω : T Xreg → Ω 1Xreg . Then the structure sheaf O Xreg admits a natural Poisson structure defined by {f, g} = ω(ω −1 (df ), ω −1 (dg)) for any sections f, g of O Xreg . Namely, the pairingis a skew C-bilinear form satisfying the Leibniz rule {f g, h} = f {g, h} + g{f, h} and the Jacobi identity {f, {g, h}} + {g, {h, f }} + {h, {g, f }} = 0 for any sections f, g and h of O Xreg .