Abstract not found

I will discuss in this paper number theoretic problems which are of combinatorial nature. I certainly do not claim to cover the field completely and the paper will be biased heavily towards problems considered by me and my collaborators. Combinatorial methods have often been used successfully in number theory (e.g. sieve methods), but here we will try to restrict ourselves to problems which themselves have a combinatorial flavor. I have written several papers in recent years on such problems and in order to avoid making this paper too long, wherever possible, will discuss either problems not mentioned in the earlier papers or problems where some progress has been made since these papers were written. Before starting the discussion of our problems I give a few of the principal papers where similar problems were discussed and where further literature can be found.

We prove a theorem on the extension of holomorphic sections of powers of adjoint bundles from submanifolds of complex codimension 1 having non-trivial normal bundle. The first such result, due to Takayama, considers the case where the canonical bundle is twisted by a line bundle that is a sum of a big and nef line bundle and a Q-divisor that has kawamata log terminal singularites on the submanifold from which extension occurs. In this paper we weaken the positivity assumptions on the twisting line bundle to what we believe to be the minimal positivity hypotheses. The main new idea is an L 2 extension theorem of Ohsawa-Takegoshi type, in which twisted canonical sections are extended from submanifolds with non-trivial normal bundle. 1.

In this paper, we present several doubly infinite families of linear projective codes with two-, three- and four distinct non-zero Hamming weights together with the frequency distributions of their weights. The codes have been defined as linear spaces of coordinate vectors of points on certain projective sets described in terms of Hermitian and quadratic forms-non-degenerate and singular- in projective spaces. The weight-distributions have been derived by considering the geometry of intersections of projective sets by hyperplanes in relevant projective spaces. Results from Bose and Chakravarti (1966)

Let X be a compact complex algebraic manifold, E → X a holomorphic line bundle, and Z ⊂ X a smooth codimension-1 submanifold. The main goal of this paper is to establish sufficient conditions for extending sections of the pluri-adjoint bundles m(KZ + E|Z) from Z to X.