TY - JOUR

T1 - A classification of SNC log symplectic structures on blow-up of projective spaces

AU - Okumura, Katsuhiko

N1 - Publisher Copyright:
© 2020, Springer Nature B.V.

PY - 2020/10/1

Y1 - 2020/10/1

N2 - 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.

AB - 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.

KW - Degeneracy loci

KW - Fano variety

KW - Holomorphic Poisson structure

KW - Log symplectic form

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U2 - 10.1007/s11005-020-01309-6

DO - 10.1007/s11005-020-01309-6

M3 - Article

AN - SCOPUS:85089256925

VL - 110

SP - 2763

EP - 2778

JO - Letters in Mathematical Physics

JF - Letters in Mathematical Physics

SN - 0377-9017

IS - 10

ER -