ABSTRACT
Two reducible germs of holomorphic varieties are weakly equivalent precisely when their irreducible components are correspondingly weakly equivalent in pairs, since removing the singular locus from a variety leaves the irreducible components as disjoint varieties, so it is only necessary to prove the theorem in the special case of irreducible germs of varieties. On the one hand, any two irreducible germs of holomorphic varieties having the same normalizations are weakly equivalent to each other. On the other hand, if two irreducible germs of holomorphic varieties are weakly equivalent, then their normalizations are also weakly equivalent, so to conclude the proof it is sufficient just to show that any two normal germs that are weakly equivalent are actually equivalent germs of varieties. The function theory on a variety and on its normalization are very closely related, indeed are identical if only weakly holomorphic or meromorphic functions are considered, so it is really enough to restrict to normal varieties.
