Complex Algebraic Geometry

The talk is based on the joint work with Boris Doubrov. First we will describe a new rather effective procedure of symplectification for the problem of local equivalence of nonholonomic vector distributions. The starting point of this procedure is to lift a distribution $D$ to a special odd-dimensional submanifold $W_D$ of the cotangent bundle, foliated by the characteristic curves (the abnormal extremals of the distribution $D$). In particular, if $D$ is a rank $2$ distribution then the submanifolds $W_D$ is nothing but the annihilator of the square of $D$, denoted by $(D^2)^\perp$, while if $D$ is a distribution of odd rank it is the annihilator of $D$ itself. The dynamics of the lifting (to $W_D$) of the distribution $D$ along the characteristic curves (of $W_D$) is described by certain curves of flags of isotropic and coisotropic subspaces in a linear symplectic space. So, the problem of equivalence of distributions can be essentially reduced to the differential geometry of such curves: symplectic invariants of these curves automatically produce invariants of the distribution $D$ itself and the canonical frame bundles, associated with such curves can be in many cases effectively used for the construction of the canonical frames of the distributions $D$ itself on certain fiber bundles over $W_D$. In this way we succeeded to construct the canonical frames for germs of rank 2 distributions in $\mathbb R^n$ with $n>5$ and of rank 3 distributions in $\mathbb R^n$ with $n>6$ from certain generic classes. The first case generalizes the classical work of E. Cartan (1910) on rank $2$ distributions in $\mathbb R^5$. The second case is also new: the only rank 3 distributions with functional invatiants, treated before, were rank $3$ distributions in $\mathbb R^5$ (Cartan, 1910) and in $\mathbb R^6$ (N. Tanaka school and independently R. Bryant in 70th). In all these cases the most symmetric models will be given as well.