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Poisson (co)homology and isolated singularities. (English) Zbl 1113.17009
Let $\mathbb{F}$ be a field of characteristic $0$ and $\mathcal{A}=\mathbb{F}[x,y,z]$. Given any $\varphi\in \mathcal{A}$, the relations $\{x,y\}_{\varphi}=\frac{\partial \varphi}{\partial z}$, $\{y,z\}_{\varphi}=\frac{\partial \varphi}{\partial x}$, $\{z,x\}_{\varphi}=\frac{\partial \varphi}{\partial y}$ define a Poisson bracket on $\mathcal{A}$, which admits $\varphi$ as a Casimir function. Therefore, this bracket induces Poisson structures both on the affine three space $F^{3}$ and the surface $\{\varphi=0\}\subset F^{3}$. Suppose that $\varphi$ is a weighted homogeneous polynomial such that the surface $\{\varphi=0\}$ has an isolated singularity at the origin. The author computes the Poisson cohomology and homology modules of the Poisson structures on $F^{3}$ and $\{\varphi=0\}$ in this case. The paper also contains clear explanations of each of the concepts mentioned.

17B63Poisson algebras
14F99Homology and cohomology theory (algebraic geometry)
17B56Cohomology of Lie (super)algebras
Full Text: DOI arXiv
[1] Alev, J.; Lambre, T.: Comparaison de l’homologie de Hochschild et de l’homologie de Poisson pour une déformation des surfaces de Klein. Algebra and operator theory, tashkent, 1997, 25-38 (1998)
[2] Brylinsky, J. -L.: A differential complex for Poisson manifolds. J. differential geom. 28, No. 1, 93-114 (1988)
[3] Cox, D.; Little, J.; O’shea, D.: Using algebraic geometry. Grad texts in math. 185 (1998)
[4] Dufour, J. -P.; Zung, N. T.: Poisson structures and their normal forms. Progr. math. 242 (2005) · Zbl 1082.53078
[5] Eisenbud, D.: Commutative algebra, with a view toward algebraic geometry. Grad texts in math. 150 (1995) · Zbl 0819.13001
[6] Ginzburg, V. L.; Lu, J. -H.: Poisson cohomology of Morita-equivalent Poisson manifolds. Int. math. Res. not. (10), 199-205 (1992) · Zbl 0783.58026
[7] Ginzburg, V. L.; Weinstein, A.: Lie -- Poisson structure on some Poisson Lie groups. J. amer. Math. soc. 5, No. 2, 445-453 (1992) · Zbl 0766.58018
[8] Haraki, A.: Quadratisation de certaines structures de Poisson. J. London math. Soc. (2) 56, No. 2, 384-394 (1997) · Zbl 0988.37068
[9] Huebschmann, J.: Poisson cohomology and quantization. J. reine angew. Math. 408, 57-113 (1990) · Zbl 0699.53037
[10] C. Laurent-Gengoux, A. Pichereau, P. Vanhaecke, An Invitation to Poisson Structures, in preparation · Zbl 1284.53001
[11] Lichnerowicz, A.: LES variétés de Poisson et leurs algèbres de Lie associées. J. differential geom. 12, No. 2, 253-300 (1977) · Zbl 0405.53024
[12] Milnor, J.: Singular points of complex hypersurfaces. Ann. of math. Stud. 61 (1968) · Zbl 0184.48405
[13] Monnier, P.: Poisson cohomology in dimension two. Israel J. Math. 129, 189-207 (2002) · Zbl 1077.17018
[14] Nakanishi, N.: Poisson cohomology of plane quadratic Poisson structures. Publ. res. Inst. math. Sci. 33, No. 1, 73-89 (1997) · Zbl 0970.53042
[15] Roger, C.; Elgaliou, M.; Tihami, A.: Une cohomologie pour LES algèbres de Lie de Poisson homogènes. Publ. dép. Math. nouvelle sér. 1990, 1-26 (1990) · Zbl 0841.58027
[16] Roger, C.; Vanhaecke, P.: Poisson cohomology of the affine plane. J. algebra 251, No. 1, 448-460 (2002) · Zbl 0998.17023
[17] Stanley, R. P.: Invariants of finite groups and their applications to combinatorics. Bull. amer. Math. soc. (N.S.) 1, No. 3, 475-511 (1979) · Zbl 0497.20002
[18] Sturmfels, B.: Algorithms in invariant theory. Texts monogr. Symbolic computat. (1993)
[19] Vaisman, I.: Lectures on the geometry of Poisson manifolds. Progr. math. 118 (1994) · Zbl 0810.53019
[20] Den Bergh, M. Van: Noncommutative homology of some three-dimensional quantum spaces. Proceedings of conference on algebraic geometry and ring theory in honor of michael Artin, part III, Antwerp, 1992, vol. 8 8, 213-230 (1994)
[21] Vanhaecke, P.: Integrable systems in the realm of algebraic geometry. Lecture notes in math. 1638 (2001) · Zbl 0997.37032
[22] Xu, P.: Poisson cohomology of regular Poisson manifolds. Ann. inst. Fourier (Grenoble) 42, No. 4, 967-988 (1992) · Zbl 0759.58020
[23] Xu, P.: Gerstenhaber algebras and BV-algebras in Poisson geometry. Comm. math. Phys. 200, No. 3, 545-560 (1999) · Zbl 0941.17016