Regularity in several variables

Abstract

The central topic of this chapter is the notion of regularity in several variables. For an algebraic integrable connection ∇ on the complement of a divisor Z in an algebraic varietyXthe notion of regularity along Z may be defined, or characterized, in at least four different algebraic ways:

• a) in terms of the iterated action of anysinglevector fieldDgenerically transversal toZ:the order of the poles occurring in the action ofD“is at most n +constant,

• b) by the fact that the logarithmic differential operators of increasing order act with poles of bounded order at the generic point of Z,

• c) via the classical notion of regularity in one variable, applied to the restriction of V to sufficiently many smooth curves inXintersecting Z transversally,

• d) by the existence of an extension with logarithmic poles along Z.