Analytic Sets and Complex Spaces

Abstract

Here we study the properties of singular analytic sets and complex spaces. The central theme of the first half of this chapter is “Oka’s Second Coherence Theorem”, claiming the coherence of a geometric ideal sheaf (the ideal sheaf of an analytic set). By making use of it, the subset of singular points of an analytic set is proved to be an analytic subset of lower dimension. In the latter half, the notion of a complex space is introduced. Oka’s normalization theorem, which reduces a singular point to a normal one with better property, and “Oka’s Third Coherence Theorem” claiming the coherence of the normalization sheaf are proved.

Historical Supplements

We refer the readers to Chap. 9 for terming Oka’s Second and Third Coherence Theorems. In writing here, the author feels still the difficulty of the proof of Oka’s Third Coherence Theorem, more difficult than that of Levi’s Problem (Hartogs’ Inverse Problem) dealt with in the next chapter.

In the comment to Oka VIII of Oka’s Collected Works [67], H. Cartan begins with writing

Ce Mémoire VIII, de lecture très difficile, est ...

The proof presented here is due to Grauert–Remmert [28], which is considerably simplified, but still requires a lot to catch the proof. This proof is already introduced in R. Narasimhan [48] p. 121 (1966), so it should have been known rather a long time before.

K. Oka invented the notion and the theory of “ideals or modules with undetermined domains”, i.e., of “Coherent Sheaf”, in order to solve Levi’s Problem (Hartogs’ Inverse Problem) even for ramified covering domains over ${\textbf{C}}^n$. But it did not go well, and he restricted himself to the case of unramified domains (i.e., Riemann domains) and wrote up Oka IX, giving a complete solution of Levi’s Problem (Hartogs’ Inverse Problem), which was then a big problem unsolved for a long time (cf. the next chapter). Therefore, after all, Oka’s Second and Third Coherence Theorems that are essential in the singular (ramified) case, were not used there.

On the other hand, taking a look into H. Cartan’s paper [9] (1944), one may find that he seems to have been concerned with problems of Cousin type, and the name “Cousin” appears quite frequently, but no mention of Levi’s Problem (Hartogs’ Inverse Problem). Therefore there was a difference in what they were looking for there.

The above facts may be interesting in view of mathematical developments.

Exercises

• 1. Let $U \subset {\textbf{C}}^n$ be an open set, and let $f \in {\mathscr {O}}(U)$. Let $X=\{f=0\}$ and let $a \in X$. Assume that $\mathscr {I}{\langle }X {\rangle }_a={\mathscr {O}}_{{\textbf{C}}^n, a} \cdot \underline{f}_a$. Show that there is a neighborhood V of a such that $\varSigma (X)\cap V=\{f=df=0\}\cap V$.

• 2. Prove Proposition 6.5.19.

• 3. (Analytic Sard’s Theorem). Let $\varOmega \subset {\textbf{C}}^n$ be a domain and let $f \in {\mathscr {O}}(\varOmega )$. Then, show the existence of an at most countable subset $Z \subset {\textbf{C}}$ such that for all $w \in {\textbf{C}}{\setminus } Z$,

  $$ df(z) \not =0, \quad ^\forall z \in f^{-1} w. $$

• 4. Let $X=\{(z,w)\in {\textbf{C}}^2; \, w^2=z^3\}$. Then, ${{\varSigma }}(X)=\{0\}$ and X is not normal at 0, as shown in Example 6.10.8. Find the normalization of X.

• 5. Let $X=\{(u,v,w) \in {\textbf{C}}^3;\, w^2=uv\}$. Show that $\varSigma (X)=\{0\}$ and that X is normal.

  (Hint: Consider a holomorphic map $\pi : (u,v) \in {\textbf{C}}^2 \rightarrow (u^2,v^2, uv) \in X$ and the pull-back $\pi ^* f$ for f with $\underline{f}_0 \in \tilde{{\mathscr {O}}}_{X,0}$.)

• 6. Let $U \subset {\textbf{C}}^n$ be an open set, and let $Y \subset X \subset U$ be analytic subsets. Let $f:X \rightarrow {\textbf{C}}$ be a continuous function which is weakly holomorphic on X. Show that the restriction $f|_Y$ is weakly holomorphic on Y. (This is not trivial when $Y \subset \varSigma (X)$.)

• 7. Let X be a normal complex space, and let Y be a thin analytic subset of X. Let $f:X {\setminus } Y \rightarrow {\textbf{C}}$ be a holomorphic function such that f is locally bounded near every point of Y. Show that f extends holomorphically on X.

• 8. In ${\textbf{C}}^3$ we define a subset A given by points $(z_1,z_2,z_3)$ as follows:

  $$ z_1=u,\quad z_2=uv, \quad z_3=uve^v $$

  for $(u,v) \in {\textbf{C}}^2$. Show that $A \cap \{z_1 \not =0\}$ is an analytic subset of $\{z_1 \not =0\}$, but that A is not an analytic subset in any neighborhood of $0 \in {\textbf{C}}^3$.

  (Hint: Show that if $\underline{f}_0 \in {\mathscr {O}}_{3,0}$ satisfies $\underline{f}_0|_{\underline{A}_0}=0$, then $\underline{f}_0=0$.)