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Reports of the Academy of Sciences of the USSR
1960. Volume 131, No. 5
MATHEMATICS
J. P. BENZECRI and F. LE ROY (J. P. BENZECRI et F. LE ROY)
ON SINGULARITIES OF ANALYTIC FIELDS
(Presented by Academician P. S. Aleksandrov, 5 I 1960)
I. Notation and definitions. Let \((F,V,p)\) be an analytic complex skew product (\(p\) denotes the projection onto the base \(V\)). An analytic subset \(S\) in \(F\) is said to define a holomorphic section over \(A\) (an open subset of \(V\)) if \(p\) is an isomorphism between \(A\) and \(p^{-1}(A)\cap S\). It is said that \(S\) defines an analytic over \(A\) section if there exists an open subset \(A'\), everywhere dense in \(A\), over which \(S\) defines a holomorphic section. It is said that the analytic section over \(A\) defined by means of \(S\) is meromorphic if \(S\cap p^{-1}(A)\) is a closed subset of \(p^{-1}(A)\). A point \(a\in A\) is called regular for the section \(S\) if there exists an open neighborhood \(O_a\) of the point \(a\) over which \(S\) defines a holomorphic section. A nonregular point \(a\) is called singular. A singular point which has no neighborhood \(O\) over which \(S\) would define a meromorphic section is called essentially singular.
Let \((E,V,p)\) be an analytic complex vector skew product (whose fiber is the vector space \(C^n\)); let \(P(E)\) be the projective skew product associated with \(E\). We define a 1-field over \(A\subset V\) as an analytic section \(\Gamma\) of the skew product \(P(E)\) over \(A\). \(S(\Gamma)\) is that part of \(E\) which is the closure of the preimage of \(\Gamma\) in \(E\).
It is said that a section \(s\) of the skew product \(E\) is included in \(\Gamma\) if
\(s\subset S(\Gamma)\).
The union of all holomorphic sections included in \(\Gamma\) (defined over some open subset of the space \(V\)) is denoted by \(I(\Gamma)\).
\(\mathcal O(V)\) denotes the sheaf of germs of functions holomorphic on \(V\), \(\mathcal O_x(V)\) the fiber of \(\mathcal O(V)\) at the point \(x\in V\). \(\mathcal F(I)\) is the sheaf of germs of holomorphic sections of \(E\) included in \(\Gamma\); \(\mathcal F(I)\) is a sheaf of modules over the sheaf of rings \(\mathcal O(V)\); \(\mathcal F_x(I)\) is the fiber of \(\mathcal F(I)\) over \(x\). \(f_x\) (respectively \(s_x\)) denotes the germ at \(x\) of the function \(f\) (respectively of the section \(s\)).
Since the results obtained by us follow from a local study, we shall assume that \(V\) is an open subset of \(C^p\) and that \(E=V\times C^n\) \((p\geq 2,\ n\geq 2)\).
A section \(s\) of the skew product \(E\) is defined by means of \(n\) analytic functions \(X_1,\ldots,X_n\).
II. Main results.
Proposition 1. Let \(s\) be a holomorphic section of \(E\) over \(A\), not identically equal to zero. Then there exists one and only one meromorphic over \(A\) 1-field \(\Gamma\) such that \(s\subset S(\Gamma)\).
Proposition 2. Let \(\Gamma\) be a meromorphic over \(A\) 1-field and let \(a\in A\). Then there exists such a neighborhood \(O_a\) of the point \(a\) that over \(O_a\) a holomorphic section \(s\) in \(E\) is defined with the following properties:
a) \(s\subset S(\Gamma)\);
b) Let \(s_x \in \mathcal F(I)\) and \(x \in A\). Then there exists \(\gamma_x \in \mathcal O_x(V)\) such that
\(s_x=\gamma_x \tilde s_x\);
c) the set of zeros of \(s\) has dimension \(\leq p-2\).
Proposition 3. The existence and commutativity of the following diagram are proved:
\[ \begin{gathered} \mathcal F(I)\ \underset{\mathfrak D}{\stackrel{\mathfrak D^{-1}}{\rightleftarrows}}\ \mathcal F(F) \\ \\[-2mm] \begin{array}{ccc} & & \\ \mathcal F(I) & & \mathcal F(F)\\ \downarrow b_1 & \searrow q & \downarrow b_F\\ I & \xleftarrow{\;D\;} & F \end{array} \end{gathered} \]
a) \(b_1\) is the mapping that transforms the germ \(s_x\) at \(x\) of a section \(s\) into \(s(x)\);
b) \(F\) is the quotient sheaf of \(\mathcal F(I)\) by the following equivalence relation: \(q(s_x)=q(\tilde s_x)\), if \(s_x=g_x\tilde s_x\) and \(g(x)=1\) \((g_x\in\mathcal O_x(V))\).
It turns out that \(F\) has the structure of a vector analytic oblique product over \(V\) with fiber \(C\).
c) \(D\) is completely determined by the relation \(D\circ q=b_1\);
d) \(\mathcal F(F)\) is the sheaf of germs of holomorphic sections of the oblique product \(F\);
e) \(\mathfrak D\) is the mapping of the sheaf \(\mathcal F(I)\) into \(\mathcal F(F)\) induced by the mapping \(D\).
It turns out that \(\mathfrak D\) is an isomorphism between the two sheaves and that the inverse mapping \(\mathfrak D^{-1}\) satisfies the relation
\(b_F\circ \mathfrak D^{-1}=q\) (\(b_F\) is the mapping of \(\mathcal F(F)\) into \(F\)).
Proposition 4. Let \(\Gamma\) be an analytic 1-field over \(V\). In order that \(\Gamma\) be meromorphic, it is necessary and sufficient that the set of singularities of the field have dimension \(\leq p-2\).
Suppose now that \(p=n\) and that \(\Gamma\) has an isolated singularity at \(0\).
Definition 1. Let \(\Sigma\) be an oriented sphere of dimension \(2n-1\), whose linking index with \(\{0\}\) is equal to 1. Then the 1-field \(\Gamma\) defines a mapping of \(\Sigma\) into \(P(n-1,C)\). The class of this mapping in \(\pi_{2n-1}(P(n-1,C))\simeq Z\) is an integer called the index of the singularity of \(\Gamma\) at \(0\).
Proposition 5. Let \(s\) be a section in \(E\), defined by Proposition 2; the index of the singularity at \(0\) is equal to the index of the mapping \(s\) of the base \(V\) into the fiber \(C^n\) (over the zero of the fiber).
Proposition 6. Let \(r\) be the projection of \(C^n-\{0\}\) onto its quotient space \(S(2n-1)\). Then \(r\circ s\) defines a mapping of the sphere \(\Sigma\) into \(S(2n-1)\), whose class in \(\pi_{2n-1}(S(2n-1))\simeq Z\) is equal to the index of the singularity.
It turns out that if \(E\) is regarded as a real \(2n\)-dimensional vector oblique product, then the index of \(\Gamma\) at \(0\) coincides with the Kronecker index of \(s\).
Proposition 7. The index is an integer positive number, which can take all values different from \(0\).
Example. According to Proposition 1, the field is completely determined by a section \(s\) in \(E\). Let us take as \(s\) the section defined by the following functions:
\[ X_1=x_1^{p}; \qquad X_i=x_i \quad (i=1,\ldots,n) \]
(where \(x_i\) are the coordinates in \(V\), and \(X_i\) are the coordinates of the fiber). Then the index of the singularity is equal to \(p\).
III. Proof scheme. The main point is Proposition 2. It is proved as follows. Suppose that \(X_1\) is not identically equal to 0 over \(I(\Gamma)\). Then the fractions \(\varphi_i = X_i/X_1\) \((i = 2, \ldots, n)\), computed over \(I(\Gamma)\), are meromorphic functions defining the field \(\Gamma\).
Denote by \(D_i^+ - D_i^-\) the divisor of the function \(\varphi_i\) in \(a\) (where \(D_i^+\), \(D_i^-\) are positive divisors without common divisors). Denote by \(D'_1\) the sum \(\sum D_i^-\) \((i = 2, \ldots, n)\); by \(D'_i\) \((i = 2, \ldots, n)\) the expressions \(D_i^+ + D'_1 - D_i^-\); and by \(D\) the greatest common divisor of all \(D'_i\) \((i = 1, \ldots, n)\).
Let \(\varphi\) be a holomorphic function, defined up to a holomorphic factor different from 0 in \(a\), whose divisor is equal to \(D'_1 - D\). Then \(X_1 = \varphi,\ X_i = \varphi \times \varphi_i\) \((i = 2, \ldots, n)\) are \(n\) holomorphic functions that define the section \(s\)*.
Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR
Received
24 XII 1959
REFERENCES
- H. Cartan, Séminaire de l’Ecole Norm. Sup., 1953—1954.
- H. Cartan, Colloque sur les fonctions de plusieurs variables, tenu à Bruxelles, 1953, Paris, 1953, p. 44.
- A. Weil, Introduction à l’étude des variétés Kaehlériennes, Paris, 1958.
* Definitions of the concepts occurring in the present article may be found in works (1–3).