Surveys in Differential Geometry

Volume 24 (2019)

Symplectic singularities of differentiable mappings

Pages: 117 – 172

DOI: https://dx.doi.org/10.4310/SDG.2019.v24.n1.a4

Authors

Goo Ishikawa (Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo, Hokkaido, Japan)

Stanisław Janeczko (Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland; and Faculty of Mathematics & Information Science, Warsaw University of Technology, Warsaw, Poland)

Abstract

Our purpose is to survey the recent results concerning the study of germs of singular varieties in a symplectic space. We formulate the theory of symplectic bifurcations with the symplectic group action on the reduced space and provide the complete symplectic classification of simple and unimodal singularities of planar curves. Their differential and symplectic invariants, e.g. symplectic defect and $\delta$‑invariant were distinguished and the corresponding cyclic moduli spaces were calculated. The classification problem of singular differentiable mappings to the symplectic space was considered and basic invariants for classification, symplectic codimension, symplectic isotropic codimension, symplectic multiplicity were constructed. The methods of geometric and algebraic restrictions of differential forms to singular varieties were presented and applied in symplectic classification of space curves and singular surfaces.

Keywords

singularities of differentiable mappings, symplectic invariants, algebraic restriction, parametric curves

2010 Mathematics Subject Classification

Primary 53D05. Secondary 57R42, 58A10, 58K05.

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Published 29 December 2021