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FIRST STEPS OF LOCAL SYMPLECTIC ALGEBRA Dedicated to D.B.Fuchs
 

Summary: FIRST STEPS OF LOCAL SYMPLECTIC ALGEBRA
V.I.Arnold
Dedicated to D.B.Fuchs
1. Introduction
The goal of this paper is the solution of a very special problem of symplectic singularity
theory -- that of the classification of the symplest singularities of curves in a symplectic
manifold. The interest of this problem consists in the following observation: there exist
nonobvious discrete symplectic invariants of such singularities. These invariants should
be expressed in terms of the local algebra's interaction with symplectic structure. The
study of this interaction, which I propose to call local symplectic algebra, is necessary for
the understanding of the results of the present paper, which at present seem to me to be
rather misterious.
The classical Darboux--Givental theorem claims, that the germ of a smooth submanifold
of a symplectic manifold is defined (up to a symplectomorphism) by the restriction of the
symplectic form to the tangent space of the submanifold.
In the case of a smooth curve this restriction vanishes. The results of the present paper
suggest that something nontrivial remains from the symplectic structure at the singular
points of the curve. It would be interesting to describe this ghost of the symplectic structure
in terms of the local algebra of the singularirty. In this paper such a formula is missing:
I just provide the classification of the curves with simplest singularities in a symplectic

  

Source: Arnold, Vladimir Igorevich - Steklov Mathematical Institute

 

Collections: Mathematics