Introduction to Algebraic Analysis on Complex Manifolds
Tadao Oda · Advanced studies in pure mathematics · 2018
This talk is meant to be an introduction for algebraic geometers, who are familiar with local cohomology formalism and derived categories but not much with hard analysis, to the so-called "algebraic analysis" on complex manifolds.This is also meant to be an introduction to Kashiwara's talk later in this symposium [K5], where he will explain to us the relevance to algebraic geometry of holonomic systems with regular singularities (the twenty-first problem of Hilbert).During the past ten years, there has been tremendous progress made in this field by Sato, Kashiwara, Kawai, Malgrange, Ramis and Mebkhout among others.The formulation, modelled after the idea and formalism of Sato's hyperfunctions, is a rather familiar one to algebraic geometers once they get used to what are going on.Most of the basic results can be found in Kashiwara [Km], [Kb], [KI], [K2] and Kashiwara-Kawai [K3].Hopefully the by now voluminous literature is made a little bit more accessible to algebraic geometers when they get familiar with the formalism and typical examples found in this talk.We touch neither on the results concerning the differential operators of infinite order nor on the "microlocal" part of the theory, which in reality play very powerful roles in the proof of the results mentioned here.As a complete newcomer myself in this field, I benefited a great deal in reading Bjork [B] and Pham [P], which are good introductions also to the micro-local theory.§ 1.What is algebraic analysis?(Ll) Let X be an n-dimensional complex manifold.We denote by ~ x the sheaf of germs of holomorphic linear partial differential operators of finite order.At a point x E X with local coordinates (z" ... , zn), the stalk of ~ x at x consists of finite sums p= 'LJxCz)(ojoz)a a