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

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Summary

Introduction

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

Poisson Structures
Outline of Pym’s Proof
Numerical Properties of Fano Products
Proof of the Main Theorem
Induction
Poisson Structures on the Product of Projective Spaces
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