# A Canonical Form of Vector Control Systems by Korovin S. K., Il’in A. V., Fomichev V. V.

By Korovin S. K., Il’in A. V., Fomichev V. V.

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It follows from the deﬁnition that D is a singular symplectic curve in Y . Generically, its only singularities are transverse double points, which may occur with either the complex orientation or the opposite orientation, and complex cusps. 2. Given a symplectic branched covering f : X → Y , the manifold X inherits a natural symplectic structure ωX , canonical up to isotopy, in the cohomology class [ωX ] = f ∗ [ωY ]. The symplectic form ωX is constructed by adding to f ∗ ωY a small multiple of an exact form α with the property that, at every point of R, the restriction of α to ker(df ) is positive.

7 (Shevchishin). Any two irreducible nodal symplectic curves in CP2 of the same degree and the same genus g ≤ 4 are symplectically isotopic. 8 (Francisco). Let C be an irreducible symplectic curve of degree d and genus 0 with κ cusps and ν nodes in CP2 , and assume that κ < d. Then C is isotopic to a complex curve. In general, we cannot expect the classiﬁcation to be so simple, and there are plenty of examples of symplectic curves which are not isotopic to any complex curve. Perhaps the most widely known such examples are due to Fintushel and Stern [11], who showed that elliptic surfaces contain inﬁnite families of pairwise non-isotopic smooth symplectic curves representing a same homology class.

Geometric structure of one-dimensional normal currents. 13 12Indeed, the notion k-tangent ﬁeld is stable under arbitrary modiﬁcations of τ in a H k - negligible set, including h-dimensional surfaces of class C 1 , while this clearly not true for the notion h-tangent ﬁeld. 1). We have not included in this paper the results about general normal currents, because they are too technical. Structure of Null Sets 19 Since T is a bounded measure, it can be written as T = τ · µ where µ is a positive measure and τ is an Rd -valued density.