Invariants of CR-manifolds associated with the tangent quadric

Abstract

ABSTRACT. We construct a system of CR-invariants of a manifold generated by projective invariants of the tangent quadric. We present a description of the group of projective diffeomorphisrns of a quartic. We also estimate the degree of a rational mapping of a quartic. The description problem for subgroups of a Cremona group of bounded degree is posed.w Introduction Consider the germ M of a smooth real surface at the origin of the complex linear space of dimension N _> 2. Then in some neighborhood of the origin the germ M may be given by the system of equationsIf the vectors {grad~a(0), p = 1,..., k} are C-linearly independent, a complex linear transformation takes the equations of M to the formhere F is a smooth mapping of a neighborhood of the origin in C n x R k into R k , and F and dF vanish at Z = 0, ReW = 0. Among such manifolds one meets (k, n)-quadrics, i.e., surfaces of the formwhere ( i.e., the surface and the quadric are tangent up to the third order. In this case the (k, n)-quadric, which we denote by Q, is called the tangent quadric to the surface M. Note that the dimension of M is k + 2n, Im W = 0 is the equation of the tangent plane at the origin, W = 0 is the equation of the complex tangent plane, i.e., n is the dimension of the complex tangent plane; and (Z, Z) is the Levi form of the surface M at the origin. Let us recall some information on tangent quartics.

Biholomorphic equivalence to totally nondegenerate model CR manifolds

“…One might be somehow surprised that so far we did not talk about the weight −ρ structure equations. In fact, our trick was to retain them for our current aim of providing at least two more weighted homogeneous equations 6 . Notice that the method suggested above, can not be applied on a weight −ρ structure equation:…”

“…Next, we start the last part, namely prolongation, of Cartan's method. Accordingly, our equivalence problem to our arbitrary CR model M k converts by that to the prolonged space M k × G red of real dimension either 3 + k or 4 + k. Finding the structure equations of this new equivalence problem is easy, it is enough to add the equation dα = 0 to the above structure equations (6). In Section 6, we start utilizing the achieved results to prove Beloshapka's maximum conjecture 1.3 in CR dimension one.…”

“…According to the principles of Cartan's theory ( [25]), once we receive the final constant type structure equations of the equivalence problem to each CR model M k , we can plainly attain the structure of its symmetry Lie algebra aut CR (M k ). Computing this algebra by means of the achieved constant type structure equations (6) together with dα = 0, we realize that it is graded and without any positive part (cf. Proposition 6.1) as was the assertion of the conjecture.…”