Local symplectic algebra of quasi-homogeneous curves

Wojciech Domitrz¹

¹ Faculty of Mathematics and Information Science Warsaw University of Technology Plac Politechniki 1 00-661 Warszawa, Poland and Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 P.O. Box 21 00-956 Warszawa, Poland

Fundamenta Mathematicae, Tome 204 (2009) no. 1, pp. 57-86.

Résumé

We study the local symplectic algebra of parameterized curves introduced by V. I. Arnold. We use the method of algebraic restrictions to classify symplectic singularities of quasi-homogeneous curves. We prove that the space of algebraic restrictions of closed $2$-forms to the germ of a $\mathbb K$-analytic curve is a finite-dimensional vector space. We also show that the action of local diffeomorphisms preserving the quasi-homogeneous curve on this vector space is determined by the infinitesimal action of liftable vector fields. We apply these results to obtain a complete symplectic classification of curves with semigroups $(3,4,5)$, $(3,5,7)$, $(3,7,8)$.

Détail

DOI : 10.4064/fm204-1-4

Keywords: study local symplectic algebra parameterized curves introduced arnold method algebraic restrictions classify symplectic singularities quasi homogeneous curves prove space algebraic restrictions closed forms germ mathbb k analytic curve finite dimensional vector space action local diffeomorphisms preserving quasi homogeneous curve vector space determined infinitesimal action liftable vector fields apply these results obtain complete symplectic classification curves semigroups