This article contains a new argument which proves vanishing of the first cohomology for negative vector bundles over a complex projective variety if the rank of the bundle is smaller than the dimension of the base. Similar argument is applied to the construction of holomorphic functions on the universal covering of the complex projective variety .

The problem of defining 2D Riemannian manifolds which admit projective motions, i. e. continuous transformation groups preserving geodesics dates back to S.Lie who considered it in [1]. In more recent times, that was A.V.Aminova who has got a complete solution to this problem [3]. For the Riemannian manifolds of dimension greater than 2 the same problem has been solved by G.Fubini [4] and A.S.S...

A set of n + k points (k > 0) in projective space of dimension n is said to be an (n + k)-arc if there is no hyperplane containing any n + 1 points of the set. It is well-known that for the projective space PG(n, q), this is equivalent to a maximum distance separable linear code with symbols in the finite field GF(q), of length n + k, dimension n + 1, and distance d = k that satisfies the Singl...

We extend the notion of Iitaka dimension of a line bundle, and deene a family of dimensions indexed by weights, for vector bundles of arbitrary ranks on complex varieties. In the case of the cotangent bundle, we get a family of birational invariants, including the Kodaira dimension and the cotangent genus introduced by Sakai. We treat in detail several classes of examples, such as complete inte...