Let f be a real-valued real analytic function in several variables. Then we associate with each algebraic local cohomology class u with support in f = 0 a distribution (generalized function) ρ(u) in terms of the residue of fλ + with respect to λ at a negative integer. Then ρ constitutes a homomorphism of modules over the sheaf of analtyic functions but not over the sheaf of differential operato...

1.1. What is this course about? The foundations of differential geometry (= study of manifolds) rely on analysis in several variables as “local machinery”: many global theorems about manifolds are reduced down to statements about what happens in a local neighborhood, and then anaylsis is brought in to solve the local problem. Analogously, algebraic geometry uses commutative algebraic as its “lo...

This is the first paper in a series of three where we develope a systematic approach to the geometric algebras of multivectors and ex-tensors. The series is followed by another one where those algebraic concepts are used in a novel presentation of the differential geometry of (smooth) manifolds of arbitrary global topology. The key calcu-lational tool in our program is the euclidean geometrical...

We construct a continuous family of new homogeneous Einstein spaces with negative Ricci curvature, obtained by deforming from the quaternionic hyperbolic space of real dimension 12. We give an explicit description of this family, which is made up of Einstein solvmanifolds which share the same algebraic structure (eigenvalue type) as the rank one symmetric space HH. This deformation includes a c...

We extend the ideas of Friedman and Qin [5] to find the wall-crossing formulae for the Donaldson invariants of algebraic surfaces with pg = 0, q > 0 and anticanonical divisor −K effective, for any wall ζ with lζ = 1 4 (ζ − p1) being 0 or 1.

From the fact that the unique solution of a homogeneous linear algebraic system is the trivial one we can obtain the existence of a solution of the nonhomogeneous system. Coefficients of the systems considered are elements of the Colombeau algebra of generalized real numbers. It is worth mentioning that the algebra is not a field.

We give a summary, without proofs, of basic properties of linear algebraic groups, with particular emphasis on reductive algebraic groups.

In this paper we discuss the notion of smoothness in complex algebraic supergeometry and we prove that all affine complex algebraic supergroups are smooth. We then prove the stabilizer theorem in the algebraic context, providing some useful applications.

The work of M. S. Livšic and his collaborators in operator theory associates to a system of commuting nonselfadjoint operators an algebraic curve. Guided by the notion of rational transformation of algebraic curves, we define the notion of a rational transformation of a system of commuting nonselfadjoint operators.

Let K be a number field and OK be the ring of algebraic integers. We discuss the unique factorization of elements of OK into irreducibles and its use in solving Diophantine equations. We then proceed to prove the existence of the unique factorization of ideals of OK into prime ideals.