Intersection theory in complex analytic geometry

Analytic Residue Theory in the Non-complete Intersection Case

In previous work of the authors and their collaborators (see, e.g., Progress in Math. 114, Birkhäuser (1993)) it was shown how the equivalence of several constructions of residue currents associated to complete intersection families of (germs of) holomorphic functions in C could be profitably used to solve algebraic problems like effective versions of the Nullstellensatz. In this work, the auth...

Model Theory and Complex Geometry

Model theory is a branch of mathematical logic whose techniques have proven to be useful in several disciplines, including algebra, algebraic geometry and number theory. The last fifteen years have also seen the application of model theory to bimeromorphic geometry, which is the study of compact complex manifolds up to bimeromorphic equivalence. In this article I will try to explain why logic s...

Operator Theory and Complex Geometry

Complex Geometry and Operator Theory

One of the principal goals of spectral theory for operators is to find unitary invariants which are local relative to the spectrum. Multiplicity theory provides a complete set of such invariants for normal operators on (complex) Hilbert space. For general operators on finite-dimensional Hilbert space a nilpotent operator is attached to each point of the spectrum and these "local operators" toge...

Jet Spaces in Complex Analytic Geometry: an Exposition

In [7] Pillay described the model-theoretic significance of a result in complex analytic geometry due to Campana [1] and Fujiki [2]. These notes are an exposition of the “jet space” constructions that underly the Campana/Fujiki theorem. In particular, we discuss infinitesimal neighbourhoods as well as the sheaves of principal parts, jets, and differential operators. The material is drawn largel...