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Elliptic singularities on log symplectic manifolds and Feigin–Odesskii Poisson brackets

Part of: Singularities Surfaces and higher-dimensional varieties Symplectic geometry, contact geometry

Published online by Cambridge University Press:  13 March 2017

Brent Pym
Brent Pym*
Affiliation:
Mathematical Institute and Jesus College, University of Oxford, Woodstock Road, Oxford OX2 6GG, UK email [email protected]
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Abstract

A log symplectic manifold is a complex manifold equipped with a complex symplectic form that has simple poles on a hypersurface. The possible singularities of such a hypersurface are heavily constrained. We introduce the notion of an elliptic point of a log symplectic structure, which is a singular point at which a natural transversality condition involving the modular vector field is satisfied, and we prove a local normal form for such points that involves the simple elliptic surface singularities $\widetilde{E}_{6},\widetilde{E}_{7}$ and $\widetilde{E}_{8}$. Our main application is to the classification of Poisson brackets on Fano fourfolds. For example, we show that Feigin and Odesskii’s Poisson structures of type $q_{5,1}$ are the only log symplectic structures on projective four-space whose singular points are all elliptic.

Keywords

Poisson structureelliptic curveFano manifoldhypersurface singularitylogarithmic differential form

MSC classification

Primary: 53D17: Poisson manifolds; Poisson groupoids and algebroids 32S25: Surface and hypersurface singularities 32S65: Singularities of holomorphic vector fields and foliations 14J45: Fano varieties
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 4 , April 2017 , pp. 717 - 744
Copyright
© The Author 2017 

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References

Aleksandrov, A. G., Nonisolated Saito singularities , Mat. Sb. (N.S.) 137(179) (1988), 554–567, 576; MR 981525 (90b:32024).Google Scholar
Aleksandrov, A. G., Nonisolated hypersurface singularities , in Theory of singularities and its applications, Advances in Soviet Mathematics, vol. 1 (American Mathematical Society, Providence, RI, 1990), 211–246; MR 1089679 (92b:32039).Google Scholar
Atiyah, M. and Hitchin, N., The geometry and dynamics of magnetic monopoles, M. B. Porter Lectures (Princeton University Press, Princeton, NJ, 1988); MR 934202 (89k:53067).CrossRefGoogle Scholar
Barlet, D., Le faisceau 𝜔 X ⋅ sur un espace analytique X de dimension pure , in Fonctions de plusieurs variables complexes, III (Sém. François Norguet, 1975–1977), Lecture Notes in Mathematics, vol. 670 (Springer, Berlin, 1978), 187–204; MR 521919 (80i:32037).Google Scholar
Bartocci, C. and Macrí, E., Classification of Poisson surfaces , Commun. Contemp. Math. 7 (2005), 89–95; MR 2129789 (2005m:14064).Google Scholar
Bondal, A. I., Non-commutative deformations and Poisson brackets on projective spaces, Preprint (1993), Max-Planck Institute Report no. 93-67.Google Scholar
Bottacin, F., Poisson structures on Hilbert schemes of points of a surface and integrable systems , Manuscripta Math. 97 (1998), 517–527; MR 1660136 (99m:14003).Google Scholar
Buchsbaum, D. A. and Eisenbud, D., Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3 , Amer. J. Math. 99 (1977), 447–485; MR 0453723 (56 #11983).CrossRefGoogle Scholar
Bursztyn, H. and Radko, O., Gauge equivalence of Dirac structures and symplectic groupoids , Ann. Inst. Fourier (Grenoble) 53 (2003), 309–337; MR 1973074 (2004d:53101).Google Scholar
Castro-Jiménez, F. J., Narváez-Macarro, L. and Mond, D., Cohomology of the complement of a free divisor , Trans. Amer. Math. Soc. 348 (1996), 3037–3049; MR 1363009 (96k:32072).Google Scholar
Cavalcanti, G. R., Examples and counter-examples of log-symplectic manifolds, Preprint (2013), arXiv:1303.6420.Google Scholar
Cavalcanti, G. R. and Gualtieri, M., Stable generalized complex structures, Preprint (2015), arXiv:1503.06357.Google Scholar
Cerveau, D. and Lins Neto, A., Irreducible components of the space of holomorphic foliations of degree two in CP(n), n⩾3 , Ann. of Math. (2) 143 (1996), 577–612; MR 1394970 (97i:32039).Google Scholar
Deligne, P., Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, vol. 163 (Springer, Berlin, 1970); MR 0417174 (54 #5232).Google Scholar
Dongho, J., Logarithmic Poisson cohomology: example of calculation and application to prequantization , Ann. Fac. Sci. Toulouse Math. (6) 21 (2012), 623–650; MR 3052027.Google Scholar
Druel, S., Structures de Poisson sur les variétés algébriques de dimension 3 , Bull. Soc. Math. France 127 (1999), 229–253; MR 1708651 (2000h:14013).Google Scholar
Ellingsrud, G. and Laksov, D., The normal bundle of elliptic space curves of degree 5 , in 18th Scandinavian Congress of Mathematicians (Aarhus, 1980), Progress in Mathematics, vol. 11 (Birkhäuser, Boston, MA, 1981), 258–287; MR 633362 (83h:14006).Google Scholar
Eriksen, E. and Gustavsen, T. S., Lie–Rinehart cohomology and integrable connections on modules of rank one , J. Algebra 322 (2009), 4283–4294; MR 2558865 (2011d:13040).Google Scholar
Etingof, P. and Ginzburg, V., Noncommutative del Pezzo surfaces and Calabi–Yau algebras , J. Eur. Math. Soc. (JEMS) 12 (2010), 1371–1416; MR 2734346 (2012d:16082).Google Scholar
Feĭgin, B. L. and Odesskiĭ, A. V., Sklyanin’s elliptic algebras , Funktsional. Anal. i Prilozhen. 23 (1989), 45–54; 96; MR 1026987 (91e:16037).Google Scholar
Feĭgin, B. L. and Odesskiĭ, A. V., Vector bundles on an elliptic curve and Sklyanin algebras , in Topics in quantum groups and finite-type invariants, American Mathematical Society Translations, Series 2, vol. 185 (American Mathematical Society, Providence, RI, 1998), 65–84; MR 1736164 (2001f:14063).Google Scholar
Goto, R., Unobstructed deformations of generalized complex structures induced by C ∞ logarithmic symplectic structures and logarithmic Poisson structures , in Geometry and topology of manifolds, Springer Proceedings in Mathematics & Statistics, vol. 154, eds Futaki, A., Miyaoka, R., Tang, Z. and Zhang, W. (Springer, Tokyo, 2016).Google Scholar
Goto, R., Rozansky–Witten invariants of log symplectic manifolds , in Integrable systems, topology, and physics (Tokyo, 2000), Contemporary Mathematics, vol. 309 (American Mathematical Society, Providence, RI, 2002), 69–84; MR 1953353 (2004b:53065).Google Scholar
Graf v. Bothmer, H.-C. and Hulek, K, Geometric syzygies of elliptic normal curves and their secant varieties , Manuscripta Math. 113 (2004), 35–68; MR 2135560 (2006b:14009).Google Scholar
Gualtieri, M. and Li, S., Symplectic groupoids of log symplectic manifolds , Int. Math. Res. Not. IMRN (2014), 3022–3074; MR 3214314.Google Scholar
Gualtieri, M., Li, S., Pelayo, A. and Ratiu, T., The tropical momentum map: a classification of toric log symplectic manifolds , Math. Ann. (2016), doi:10.1007/s00208-016-1427-9.Google Scholar
Guillemin, V., Miranda, E. and Pires, A. R., Symplectic and Poisson geometry on b-manifolds , Adv. Math. 264 (2014), 864–896; MR 3250302.Google Scholar
Guillemin, V., Miranda, E., Pires, A. R. and Scott, G., Toric actions on b-symplectic manifolds , Int. Math. Res. Not. IMRN (2015), 5818–5848.Google Scholar
Gualtieri, M. and Pym, B., Poisson modules and degeneracy loci , Proc. Lond. Math. Soc. (3) 107 (2013), 627–654; MR 3100779.Google Scholar
Hartshorne, R., Stable reflexive sheaves , Math. Ann. 254 (1980), 121–176; MR 597077 (82b:14011).Google Scholar
Hulek, K., Projective geometry of elliptic curves , in Algebraic geometry—open problems (Ravello, 1982), Lecture Notes in Mathematics, vol. 997 (Springer, Berlin, 1983), 228–266; MR 714752 (85g:14035).Google Scholar
Ingalls, C., Quantizable orders over surfaces , J. Algebra 207 (1998), 616–656; MR 1644207 (99h:14045).Google Scholar
Lima, R. and Vitório Pereira, J., A characterization of diagonal Poisson structures , Bull. Lond. Math. Soc. 46 (2014), 1203–1217; MR 3291256.Google Scholar
Loray, F., Vitório Pereira, J. and Touzet, F., Foliations with trivial canonical bundle on Fano 3-folds , Math. Nachr. 286 (2013), 921–940; MR 3066408.Google Scholar
Mărcuţ, I. and Osorno Torres, B., Deformations of log-symplectic structures , J. Lond. Math. Soc. (2) 90 (2014), 197–212; MR 3245143.Google Scholar
Mărcuţ, I. and Osorno Torres, B., On cohomological obstructions for the existence of log-symplectic structures , J. Symplectic Geom. 12 (2014), 863–866; MR 3333030.Google Scholar
Nevins, T. A. and Stafford, J. T., Sklyanin algebras and Hilbert schemes of points , Adv. Math. 210 (2007), 405–478; MR 2303228 (2008f:14002).Google Scholar
Nest, R. and Tsygan, B., Formal deformations of symplectic manifolds with boundary , J. Reine Angew. Math. 481 (1996), 27–54; MR 1421945 (98f:58093).Google Scholar
Polishchuk, A., Algebraic geometry of Poisson brackets , J. Math. Sci. (N.Y.) 84 (1997), 1413–1444; Algebraic geometry, 7. MR 1465521 (98d:58065).Google Scholar
Polishchuk, A., Poisson structures and birational morphisms associated with bundles on elliptic curves , Int. Math. Res. Not. IMRN (1998), 683–703; MR 1636545 (99f:14014).Google Scholar
Radko, O., A classification of topologically stable Poisson structures on a compact surface , J. Symplectic Geom. 1 (2002), 523–542.Google Scholar
Ran, Z., Deformations of holomorphic pseudo-symplectic Poisson manifolds , Adv. Math. 304 (2017), 1156–1175.Google Scholar
Saito, K., Quasihomogene isolierte Singularitäten von Hyperflächen , Invent. Math. 14 (1971), 123–142; MR 0294699 (45 #3767).Google Scholar
Saito, K., Theory of logarithmic differential forms and logarithmic vector fields , J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), 265–291; MR 586450 (83h:32023).Google Scholar
Terao, H., Arrangements of hyperplanes and their freeness. I , J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), 293–312; MR 586451 (84i:32016a).Google Scholar
Weinstein, A., The local structure of Poisson manifolds , J. Differential Geom. 18 (1983), 523–557; MR 723816 (86i:58059).CrossRefGoogle Scholar
Weinstein, A., The modular automorphism group of a Poisson manifold , J. Geom. Phys. 23 (1997), 379–394; MR 1484598 (98k:58095).Google Scholar