Abstract
AbstractThis paper classifies Poisson structures with the reduced simple normal crossing divisor on a product of Fano varieties of Picard number 1. The characterization of even-dimensional projective spaces from the viewpoint of Poisson structures is given by Lima and Pereira. In this paper, we generalize the characterization of projective spaces to any dimension.
Highlights
Geometry of log symplectic form is well investigated, and the term log symplectic is slightly abused
In Goto’s de nition [ ], log symplectic indicates the generically symplectic Poisson strucuture with the reduced and simple normal crossing degeneracy divisor. We use this terminology following Pym’s de nition [ ]. at is, we do not suppose that the reduced degeneracy divisor is smooth or simple normal crossing. is is a reasonable de nition in terms of holomorphic Poisson structures and the eld of algebraic geometry, as the degeneracy divisors usually have singularities in the higher-dimensional case [, eorem ]
Okumura e diagonal Poisson structure on a projective space is de ned as a generically symplectic Poisson structure whose degeneracy divisor is the union of all coordinate hyperplanes
Summary
Geometry of log symplectic form is well investigated, and the term log symplectic is slightly abused. ([ ]) Let (X, Π) be a log symplectic structure with the simple normal crossing degeneracy divisor (say snc log symplectic structure) on a complex Fano variety X with cyclic Picard group of even dimension n ≥ . K. Okumura e diagonal Poisson structure on a projective space (or an a ne space) is de ned as a generically symplectic Poisson structure whose degeneracy divisor is the union of all coordinate hyperplanes. A diagonal Poisson structure on the product of projective spaces indicates that the degeneracy divisor is the union of all coordinate hyperplanes. Xini denote a homogeneous coordinate system of Xi and ci jkl ∈ C. en we can express the diagonal Poisson structure on the product of the projective spaces in the following form:. Suppose that X × X admits a SNC log symplectic structure, X is a projective space
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