# zbMATH — the first resource for mathematics

Birational equivalence in the symplectic category. (English) Zbl 0683.53033
The geometry of the Marsden-Weinstein reduced phase spaces is studied for symplectic actions of the torus $$S^ 1\times S^ 1$$. The first part of the article studies the special but typical case of “blowing up” $${\mathbb{C}}^ n$$ at 0 as a symplectic manifold with reduction by $$S^ 1\times S^ 1$$. It is shown that even if the level set has singularities the reduced phase space is nonsingular. Using tubular neighborhood techniques these constructions are globalized to general symplectic manifolds. Two related constructions - “ramified coverings” and “real blowing up” - are used to show that critical levels can be split into simple critical levels. The methods of the first part provide a normal form for simple critical points. This gives a description of the change of the symplectic structure of the reduced phase space $$M_ a$$ as a passes through critical levels.
The article motivates the notion of symplectic cobordism and the main results can be formulated: Every symplectic cobordism factors into a pair of simple cobordisms and there is a canonical symplectic model for simple cobordisms. The results can be made invariant under any compact group commuting with the $$S^ 1\times S^ 1$$ action.
Reviewer: Ch.Günther

##### MSC:
 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 57S25 Groups acting on specific manifolds
Full Text:
##### References:
 [1] [A] Atiyah, M.: Convexity and commuting Hamiltonians. Bull. Lond. Math. Soc.14, 1-15 (1982) · Zbl 0482.58013 · doi:10.1112/blms/14.1.1 [2] [D-H] Duistermaat, J.J., Heckman, G.: On the variation of cohomology of the symplectic form on the reduced phase space. Invent. Math.69, 259-268 (1982) · Zbl 0503.58015 · doi:10.1007/BF01399506 [3] [G] Gromov, M.: Pseudoholomorphic curves in symplectic manifolds. Invent. Math.82, 307-347 (1985) · Zbl 0592.53025 · doi:10.1007/BF01388806 [4] [G-N] Gotay, M., Nester, J.: Pre-symplectic manifolds and the Dirac-Bergmann theory of constraints. J. Math. Phys.19, 2388-2399 (1978) · Zbl 0418.58010 · doi:10.1063/1.523597 [5] [G-S]1 Guillemin, V., Sternberg, S.: Convexity properties of the moment map. Invent. Math.67, 491-513 (1982) · Zbl 0503.58017 · doi:10.1007/BF01398933 [6] [G-S]2 Guillemin, V., Sternberg, S.: Geometric quantization and multiplicities of group representations. Invent. Math.67, 515-538 (1982) · Zbl 0503.58018 · doi:10.1007/BF01398934 [7] [G-S]3 Guillemin, V., Sternberg, S.: Symplectic Techniques in Physics. Cambridge Univ. Press 1984 · Zbl 0576.58012 [8] [H] Heckman, G.: Thesis. University of Leiden 1980 [9] [Hi] Hirzebruch, F.: The signature of ramified coverings. Lecture given at the Summer Institute on Global Analysis, AMS Berkeley, July, 1968. Symposium in honor of K. Kodaira, D.C. Spencer (ed.). AMS Pub. Providence, 1970 [10] [K-K-S] Kazhdan, D., Kostant, B., Sternberg, S.: Hamiltonian group actions and dynamical systems of Calogero type. Commun. Pure Appl. Math.31, 481-508 (1978) · Zbl 0368.58008 · doi:10.1002/cpa.3160310405 [11] [Kir] Kirwan, F.: Cohomology of quotients in symplectic and algebraic geometry. Princeton Univ. Press, Princeton, N.J. 1984 · Zbl 0553.14020 [12] [Kod] Kodaira, K.: Complex manifolds and deformations of complex structures. Berlin-Heidelberg-New York: Springer 1986 [13] [Kos] Kostant, D.: A formula for the multiplicity of a weight. Trans. Am. Math. Soc.93, 53-73 (1959) · Zbl 0131.27201 · doi:10.1090/S0002-9947-1959-0109192-6 [14] [M]1 MacDuff, D.: Examples of simply connected non-Kahlerian manifolds. J. Differ. Geom.20, 267-277 (1984) [15] [M]2 MacDuff, D.: Examples of symplectic structures. Preprint, SUNY Stony Brook 1986 · Zbl 0626.53025 [16] [Mel] Melrose, R.: Analysis on manifolds with corners. MIT 1988 [17] [M-F] Mumford, D., Fogarty, J.: Geometric invariant theory. Berlin-Heidelberg-New York: Springer 1982 · Zbl 0504.14008 [18] [N] Ness, L.: A stratification of the null cone via the moment map. Am. J. Math.106, 1281-1325 (1984) · Zbl 0604.14006 · doi:10.2307/2374395 [19] [S] Sternberg, S.: On minimal coupling and the symplectic mechanics of a classical particle in the presence of a Yang-Mills field., Proc. Natl. Acad. Sci. USA74, 5253-5254 (1977) · Zbl 0765.58010 · doi:10.1073/pnas.74.12.5253 [20] [W]1 Weinstein, A.: Lectures on Symplectic manifolds. CBMS Reg. Conf. Ser. Math., Vol. 29. AMS Providence, R.I. 1977 · Zbl 0406.53031 [21] [W]2 Weinstein, A.: Fat bundles and symplectic manifolds. Adv. Math.37, 239-250 (1980) · Zbl 0449.53035 · doi:10.1016/0001-8708(80)90035-3 [22] [Z] Zhelobenko, D.P.: Compact Lie groups and their representations. AMS Transl., Vol. 40. AMS, Providence, R.I. 1972
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.