A classification of SNC log symplectic structures on blow-up of projective spaces
- PDF / 318,563 Bytes
- 16 Pages / 439.37 x 666.142 pts Page_size
- 97 Downloads / 176 Views
A classification of SNC log symplectic structures on blow-up of projective spaces Katsuhiko Okumura1 Received: 28 August 2019 / Revised: 20 February 2020 / Accepted: 19 June 2020 / Published online: 10 August 2020 © Springer Nature B.V. 2020
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 ≥ 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. Keywords Holomorphic Poisson structure · Log symplectic form · Fano variety · Degeneracy loci Mathematics Subject Classification 22E27
1 Introduction Poisson structures on projective surfaces have been well investigated. References [1,3] are examples of previous work that report projective Poisson surfaces. But in higher dimension, we know little even on a simple variety like a projective space. One of the
B 1
Katsuhiko Okumura [email protected] Department of Mathematics, Waseda University, Tokyo, Japan
123
2764
K. Okumura
main reasons is complexity of degeneracy. It is known that degeneracy divisor must have singularities when the variety is a Fano variety of dimension greater than 4 [2]. Poisson structures are often regarded as symplectic structures that admit degeneracy. So degeneracy locus is one of the most important information in Poisson geometry. If a Poisson structure is generically symplectic, that is, on even-dimensional variety and with maximum rank, ensures that degeneracy locus becomes divisorial. By the definition, the degeneracy divisor is an anti-canonical divisor when the Poisson structure is generically symplectic. Therefore, in the perspective of birational geometry of the pair, geometry of generically symplectic Poisson structures is geometry of log Calabi–Yau pair. Recently, some papers classify generically symplectic Poisson structures with a reduced degeneracy divisor on Fano varieties. We can identify it with a log symplectic form that is a closed logarithmic 2-form. In this paper, we call a generically symplectic Poisson structure with a reduced degeneracy divisor a log symplectic structure. Classification o
Data Loading...
