In this paper we announce results, basically topological, on nonsingular complex projective varieties of special type, namely, on manifolds which can be defined by a system of equations, the number of which is one larger than the codimension (observing the natural regularity condition). Formulas are obtained for the Euler characteristic of such varieties, and in the case of codimension 2, for the Todd genus and for the signature; only the degrees of the equations and the degree of the variety appear in the formulas. Three low dimensional examples of varieties of the type considered are obtained using the so-called determinantal locus. In dimensions 2 and 3 the condition for a variety being a determinantal locus is given by a simple inequality, in which the degree of the equations and the degree of the variety appear. It turns out further that if the dimension of a variety of the type considered is not less than its codimension and is greater than 3, then it is a regular complete intersection.