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

Katsuhiko Okumura*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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 2 n≥ 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.

Original languageEnglish
Pages (from-to)2763-2778
Number of pages16
JournalLetters in Mathematical Physics
Issue number10
Publication statusPublished - 2020 Oct 1


  • Degeneracy loci
  • Fano variety
  • Holomorphic Poisson structure
  • Log symplectic form

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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