A classification of SNC log symplectic structures on blow-up of projective spaces

Abstract

It is commonly recognized that the classfication of Poisson manifold is a major problem. From the viewpoint of algebraic geometry, holomorphic projective Poisson manifold is the most important target. Poisson structures on the higher dimensional projective varieties was first studied by Lima and Pereira (Lond Math Soc 46(6):1203–1217, 2014). They proved that any Poisson structures with the reduced and simple normal crossing degeneracy divisor, we call SNC log symplectic structure, on the $2n\ge 4$ dimensional Fano variety with the cyclic Picard group must be a diagonal Poisson structure on the projective space. However, it remains to be elucidated when the Picard rank of the variety is greater or equals to 2. Here, we studied SNC log symplectic structures on blow-up of a projective space along a linear subspace, whose Picard rank equals to 2. Using Pym’s method, we have found that there are conditions on the irreducible decomposition of the degeneracy divisor and applying Polishchuk’s study Polishchuk (J Math Sci 84(5):1413–1444, 1997), we concretely described the Poisson structures corresponding to each classification result.