# On modifications and exceptional analytic sets

*1962*

#### Translator’s note

*This page is a translation into English of the following:*

Grauert, H. “Über Modifikationen und exzeptionelle analytische Mengen.” *Math. Ann.* **146** (1962), 331–368. eudml.org/doc/160940.

*The translator (Tim Hosgood) takes full responsibility for any errors introduced, and claims no rights to any of the mathematical content herein.*

Version: `ea4600c`

The term “modification” first appeared in a 1951 publication [1] by H. Behnke and K. Stein.
The authors used it to refer to a process that allows a given complex space to be modified.
If *modification* of

As already demonstrated in [1], modifications can be very pathological.
The interest therefore turned towards special classes of modifications.
In [12], H. Hopf considered so-called “

This present work deals with the following question.
Let

If such a *exceptional analytic set* in *collapsed*” to a point.

In general, such a

We now give an overview of the present work.
In §1 we study the concepts of *pseudoconvexity* and *holomorphic convexity* on complex spaces.
The reduction theory of Remmert then leads, in §2, to the first general theorem concerning exceptional analytic sets

— It should also be mentioned that, using the main results of §4, we construct a complex space

X is connected, compact, and of dimension2 ;X is normal, and has only one non-regular points;- there exist two analytically and algebraically independent meromorphic functions on
X ; and X is not an algebraic variety (neither in the projective sense nor the more general sense of Weil).^{1}

In contrast, as is well known, Kodaira and Chow [4] have shown that every compact,

# 1 Complex spaces, pseudoconvexivity

## 1.1 —

Complex spaces are defined as in [10].
We always assume that they are reduced: their local rings contain no nilpotent elements.
If

We always denote by

Now^{2} let *regular map* at

We say that a map *biholomorphic* if it is a bijection that is regular at every point

Let ^{3}

Of course,

*Proof*. To prove (1), we may assume that

- The functions
f_1,\ldots,f_r are holomorphic onW , and vanish onX\cap W ; \hat{G}=\{z\in W\mid f_v(z)=0\text{ for }v=1,2\ldots,r\} is ad(x) -dimensional analytic subset ofW that contains no singularities, and which is mapped to a domain in\mathbb{C}^{d(x)} under some biholomorphic map\tau .

Now let

To prove the second claim of (1), let

By the definition of a complex space, for every point

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Some of the results of the present work were discovered in 1959, and published in [7]. There are, however, some errors in [7]: in Theorem 1, it should, of course, read “[…] such that

G is strongly pseudoconvex andA is the maximal compact analytic subset ofG ”; furthermore, the criterion in Theorem 2 is only sufficient (see §3.8); Theorem 3 is only proven in the present work in the case where the normal bundleN(A) is weakly negative. — The author has already presented, several times, previously, the example of the complex spaceX , and, since then, Hironaka has found more interesting examples of complex spaces of this type.↩︎A subscript

x always denotes the stalk of the sheaf at the pointx . Ifs is a section, thens_x denotes its value atx . Holomorphic functions and sections in{\mathscr{O}} are always considered to be the same thing. — IfF is a complex-analytic vector bundle, then\underline{F} always denotes the sheaf of germs of locally holomorphic sections inF .↩︎This statement and its proof were communicated to me by A. Andreotti.↩︎