A characterization of diagonal Poisson structures

Abstract

Abstract. The degeneracy locus of a generically symplectic Poisson structure on a Fano manifold is always a singular hypersurface. We prove that there exists just one family of generically symplectic Poisson structures in Fano manifold with cyclic Picard group having a reduced simple normal crossing degeneracy locus.

A Bogomolov Unobstructedness Theorem for Log-Symplectic Manifolds in General Position

“…(X, ) is simply log-symplectic if moreover D( ) has simple normal crossings, i.e., is a transverse union of smooth components. We note that Lima and Pereira [11] have shown that if (X, ) is simply log-symplectic and X is a Fano manifold of dimension 4 or more with cyclic Picard group, then X is P 2n with a standard (toric) Poisson structure = a i j x i x j ∂ / ∂ x i ∂ / ∂ x j , and x i homogeneous coordinates (and consequently deformations of (X, ) are of the same kind, at least set-theoretically).…”

Singular foliations with trivial canonical class

“…We were dragged into the subject by a desire to better understand previous results, most notably [16] and [53], which we recall below. Further motivation comes from the study of holomorphic Poisson manifolds, see [48,35].…”

Mixed Hodge structures in log symplectic geometry

“…The following types of objects have been studied (in greater generality) by several authors; they were first defined by Goto [Got]. More recently, Gualtieri-Pym [GP13], and Pym [Pym17] have studied log symplectic pairs, as have Lima-Pereira [LP14], and Ran [Ran17]. We will focus on log symplectic pairs (X, Y ) where Y is a simple normal crossings divisor, but we note that this is not required in many of the works that we have mentioned.…”