Complex spaces and nonstandard schemes

Abstract

Nonstandard mathematics furnishes a remarkable connexion between analytic and algebraic geometry. We describe this interplay for the most basic notions like complex spaces/algebraic schemes, generic points, differential forms etc. We obtain -by this point of view -in particular new results on the prime spectrum of a Stein algebra.2000 Mathematics Subject Classification 32C15, 14A15 (primary); 32E10, 26E35, 58AQ10 (secondary)

Bounded Polynomials and Holomorphic Mappings Between Convex Subrings of *

“…. , T n ] → O(C n ) is surjective and its kernel is given by b C-infinitesimal polynomials, see [6]. Based on the previous results, we expect that the ring of ( b C, F)-bounded polynomials and the monoid of ( b C, F)-naturals will be the key instruments to boost the theory of holomorphic generalized functions.…”

“…We note that the proofs are not a direct transposition of those in [6] and substantial modifications are needed. The main reason is that a hyperfinite product of elements in F of length in F is not in general in F. This fact constitutes one of the fundamental disparities between b C and F, such that F b C. §2.…”

Topologizing Rings of Map Germs: An Order Theoretic Analysis of Germs via Nonstandard Methods

“…Our constructions rely critically on nonstandard methods. Given the nonstandard work on germs stretching from the 1960's (see Robinson,[26]) up to the present day (see eg., Kalfallah and Kosarew, [16]), it's rather curious that nonstandard investigations of germs of general functions (eg., beyond those with analytic type rigidities) has not occurred. Further, note that those constructions of nontrivial topologies on germs that do exist, ie., on analytic germs, do not involve nonstandard techniques.…”

Examples of new nonstandard hulls of topological vector spaces

“…algebra of entire functions on C n , see [11], and for n = 1, the Fnonstandard hull of C[T ] is the nonstandard counterpart of Colombeau's generalized holomorphic functions over C; see [13]. The proof is based on the existence of global holomorphic representatives of Colombeau's generalized holomorphic functions.…”