9.1 Complex spaces and modificationsB64C7FEAE2E7

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Analytic sets enter pluripotential theory as the pole sets of potentials with analytic singularities, as the zero sets of the ideals that describe these poles, and as the exceptional divisors of modifications. This appendix collects the facts about analytic sets, reduced complex spaces and modifications that are used from section 6.6 on: for multiplier ideals and Lelong numbers along divisors in section 6.6, for analytic singularities and their log resolutions in section 6.13, for currents supported on analytic sets in section 10.1, and for the non-Kähler locus in section 16.10. It can be consulted when these tools first appear. Besides results of the earlier chapters, such as the Weierstrass preparation theorem (theorem 5.5.2), only two existence theorems are used without proof: the resolution of singularities and the principalization of coherent ideals, stated as theorem 9.6.6, theorem 9.6.7.

The local theory rests on the Weierstrass division theorem. It shows that the ring of germs of holomorphic functions is Noetherian and, through Oka's theorem, that the sheaf of holomorphic functions is coherent, so that statements about one stalk extend to a neighborhood. The local parametrization theorem (theorem 9.3.4) presents an irreducible germ of an analytic set as a finite branched covering of a polydisc. From it we deduce the analytic Nullstellensatz, the basic properties of dimension, Cartan's theorem on the coherence of the ideal sheaf of an analytic set, and the decomposition of an analytic set into irreducible components. We also show that a closed real $(1,1)$-current with measure coefficients supported on a nowhere dense analytic set is a combination of currents of integration along hypersurfaces (theorem 9.3.20). One phenomenon requires care: an irreducible analytic set can have several local branches at one point. The normalization (theorem 9.4.6) separates these branches; its holomorphic functions are the weakly holomorphic functions, that is, the functions that are holomorphic on the regular locus and locally bounded near the singular locus. The cusp shows that a space with only one branch at each point need not be normal (example 9.4.10).

For modifications, explicit charts describe blowups along submanifolds and the transformation of ideals under them (proposition 9.6.2, proposition 9.6.3); together with the two existence theorems they give log resolutions of coherent ideals. Remmert's theorem, proved together with the Remmert–Stein extension theorem, shows that images of analytic sets under proper holomorphic maps, in particular the images of exceptional divisors, are analytic (theorem 9.7.2). We then prove a relative principalization theorem, which controls an ideal and its restriction to a submanifold simultaneously (theorem 9.7.5), show that proper bimeromorphic maps onto spaces with one local branch at each point have connected fibers (theorem 9.8.1), and prove that every prime divisor over a compact complex space appears on a model obtained by finitely many blowups with smooth centers (theorem 9.8.6).

The last section defines plurisubharmonic functions on reduced complex spaces by local extension to an ambient space, proves that they are constant on compact connected spaces, and shows that they are locally bounded above near analytic subsets of codimension at least two. Holomorphic curves through singular points, constructed in section 9.5, reduce these statements to the maximum principle in one variable. Historical remarks and references for this appendix are collected in subsection 25.1.21.