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SNC Log Symplectic Structures on Fano Products

Published online by Cambridge University Press:  24 February 2020

Katsuhiko Okumura*
Affiliation:
Department of Mathematics, Waseda University, Ookubo, Shinjuku-ku, Tokyo, 169-8555, Japan Email: [email protected]
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Abstract

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This 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.

Type
Article
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© Canadian Mathematical Society 2020

References

Cavalcanti, G. R., Examples and counter-examples of log-symplectic manifolds. J. Topol. 10(2017), 121. https://doi.org/10.1112/topo.12000CrossRefGoogle Scholar
Fujita, T., Classifications of projective varieties of 𝛥-genus one. Proc. Japan Acad. Ser. A Math. Soc. 58(1982), 113116.CrossRefGoogle Scholar
Fujita, T., On polarized varieties of small 𝛥-genera. Tohoku Math. J. (2) 34(1982), 319341. https://doi.org/10.2748/tmj/1178229197CrossRefGoogle Scholar
Goto, R., Rozansky-witten invariants of log symplectic manifolds. In: Integrable systems, topology, and physics (Tokyo, 2000). Comtemp. Math., 309, Amer. Math. Soc., Providence, RI, 2002, pp. 6984. https://doi.org/10.1090/conm/309/05342CrossRefGoogle Scholar
Gualtieri, M. and Li, S., Symplectic groupoids of log symplectic manifolds. Int. Math. Res. Not. IMRN 2014, no. 11, 30223074. https://doi.org/10.1093/imrn/rnt024CrossRefGoogle Scholar
Gualtieri, M. and Pym, B., Poisson modules and degeneracy loci. Proc. Lond. Math. Soc. 107(2013), 627654. https://doi.org/10.1112/plms/pds090CrossRefGoogle Scholar
Guillemin, V., Miranda, E., and Pires, A. R., Symplectic and poisson geometry on b-manifolds. Adv. Math. 264(2014), 864896. https://doi.org/10.1016/j.aim.2014.07.032CrossRefGoogle Scholar
Kobayashi, S. and Ochiai, T., Characterizations of complex projective spaces and hyperquadrics. J. Math. Kyoto Univ. 13(1973), 3147. https://doi.org/10.1215/kjm/1250523432CrossRefGoogle Scholar
Lazarsfeld, R., Positivity in algebraic geometry. Vol. 1, Springer-Verlag, Berlin, 2004. https://doi.org/10.1007/978-3-642-18808-4Google Scholar
Lima, R. and Pereira, J. V., A characterization of diagonal poisson structure. Bull. Lond. Math. Soc. 46(2014), 12031217. https://doi.org/10.1112/blms/bdu074CrossRefGoogle Scholar
Mǎcut, I. and Torres, B. O., Deformations of log-symplectic structures. J. Lond. Math. Soc. (2) 90(2014), 197212. https://doi.org/10.1112/jlms/jdu023CrossRefGoogle Scholar
O’Grady, K. G., A new six-dimensional irreducible symplectic variety. J. Algebraic Geom. 12(2003), 435505. https://doi.org/10.1090/S1056-3911-03-00323-0CrossRefGoogle Scholar
Okumura, K., A classification of SNC log symplectic structures on blow-up of projective spaces.Google Scholar
Pym, B., Constructions and classifications of projective poisson varieties. Lett. Math. Phys. 108(2017), 573632. https://doi.org/10.1007/s11005-017-0984-5CrossRefGoogle ScholarPubMed
Pym, B., Elliptic singularities on log symplectic manifolds and Feiginn–Odesskii Poisson brackets. Compos. Math. 153(2017), 717744. https://doi.org/10.1112/S0010437X16008174CrossRefGoogle Scholar
Radko, O., A classification of topologically stable poisson structures on a compact oriented surface. J. Symplectic Geom. 1(2002), 523542.CrossRefGoogle Scholar