25.1.21 Analytic spaces and modificationsF14EBE94C297

The Weierstrass division theorem and the Noetherian property of the ring of germs of holomorphic functions (proposition 9.2.2, proposition 9.2.4) follow [Chapter II, (2.3)–(2.7), pp. 80–81, DemBook]. For coherent sheaves and Oka's coherence theorem (lemma 9.2.6, theorem 9.2.7) compare [Chapter II, (3.8)–(3.22), pp. 87–90, DemBook]. Local irreducible components, local parametrization, the Nullstellensatz, dimension and Cartan's theorem on the coherence of the ideal sheaf of an analytic set are treated in [Chapter II, (4.1)–(4.31), pp. 91–100, DemBook]. The algebraic facts on finite field extensions and discriminants used in these constructions are proved in lemma 9.3.3 by linear algebra over the field of fractions, with matrices of multiplication and the trace form, and they are applied in theorem 9.3.4. In our proof of Cartan's theorem, theorem 9.3.10, a single denominator works at all nearby stalks, which is more than finite generation of each stalk.

The Nullstellensatz and the strict decrease of dimension for proper analytic subgerms of an irreducible germ (corollary 9.3.8, proposition 9.3.9) are deduced from the same local parametrization: division by the discriminant identifies each prime ideal with the ideal of its zero set, and the connectedness of the generic sheets gives the decrease of dimension. The proof of the Nullstellensatz includes the description of the radical as an intersection of prime ideals.

The support theorem for closed currents of order zero, theorem 9.3.20, is the case of bidegree $(1,1)$ of [Chapter III, (2.10)–(2.14), pp. 141–143, DemBook]. Our proof uses smooth normal coordinates, convolution in tangential directions and induction over the singular loci of the analytic set; the theorem is applied in lemma 10.5.7. In the proof of proposition 10.5.8, the pullback is corrected with the pushforward of a Kähler form, which gives the required strict positivity directly.

For universal denominators of weakly holomorphic functions and for normalization (proposition 9.4.3, theorem 9.4.6) compare [Chapter II, (7.1)–(7.15), pp. 110–116, DemBook]. The cusp and the pair of crossing lines in example 9.4.10 illustrate the difference between being normal and being unibranch. The Remmert–Stein extension theorem and Remmert's proper mapping theorem are proved together in [Chapter II, (8.7)–(8.8), pp. 118–121, DemBook]. Our proof of theorem 9.7.1, theorem 9.7.2 also treats them together, but it uses the resolution of singularities, theorem 9.6.6, to reduce to proper maps defined on manifolds, and it proves the statements on finite maps and on the dimension of hypersurfaces separately.

The resolution and principalization theorems (theorem 9.6.6, theorem 9.6.7) are the analytic versions of [Theorems 1.1 and 1.3, pp. 611 and 614; Definitions 2.1–2.2, pp. 618–619, BM08]; that the results apply to complex analytic spaces is stated on p. 611 there. The relative principalization theorem, theorem 9.7.5, follows the lemma on relative principalization in the unpublished manuscript [Lemma relative-principalization, VuXiaNakamaye]. At each step of this construction, the centers in the ambient manifold divide the ideal by exactly one exceptional factor, as required for the controlled transform of the restricted ideal. In our statement the submanifold is closed and the ambient manifold is connected, so that the global blowups and the identities between ideals hold as stated.

Theorem 9.8.1 states that a proper bimeromorphic map has connected fibers over the points at which the germ of the target is irreducible, in particular over unibranch spaces. Its proof follows the argument with complements of branches in [Appendix on unibranch spaces, XiaSampleBook]; properness is used to place the inverse image of a small neighborhood inside neighborhoods of the pieces of the fiber. The local upper bound near analytic subsets of codimension at least two in a reduced space, proposition 9.9.4, is proved by slicing with curves over a finite projection and the maximum principle for subharmonic functions. It complements the extension theorem on manifolds, corollary 9.3.18, which follows from the criterion in terms of Hausdorff measure.

The corollary on the realization of divisors, in the appendix of [XiaSampleBook] on unibranch spaces, underlies the comparison of multiplier ideals with Lelong numbers along divisors in Part I of that book. Theorem 9.8.6 proves this corollary by principalizing the successive centers and decreasing the order of the Jacobian along a fixed divisor. The definition of plurisubharmonic functions on reduced spaces by local ambient extension, definition 9.9.1, suffices for pullback by holomorphic maps, for the maximum principle on compact fibers and for the upper bound obtained by slicing with curves. On the manifolds used in Part I of that book, the descent of quasi-plurisubharmonic functions under modifications, proposition 10.3.9, is proved directly by extension across an analytic set of codimension at least two.