ABSTRACT
Dimension is really the only local invariant of a complex manifold, since clearly two germs of regular holomorphic varieties are biholomorphic precisely when they have the same dimension. That is not the case for arbitrary holomorphic varieties, though, as is quite apparent from the examples that have already been discussed, but dimension is nonetheless the basic and most useful of the invariants associated to holomorphic varieties. The algebraic interpretation of the dimension of a holomorphic subvariety can be used to derive some other simple properties and characterizations of dimension. The purely algebraic characterization of the dimension of the germ of a holomorphic variety in terms of its local ring is expressed in terms of the prime ideals of the local ring, paralleling a corresponding characterization of dimension in algebraic geometry, and rests on the notions.
