6)
An
analytic set in the theory of analytic functions
is a set that can locally be defined as the
set of common zeros of a finite number of holomorphic functions. If
is an analytic set in an open subset
of the complex
-dimensional
space
,
this means that for each point
there exist a neighbourhood
and a finite tuple of functions
,
holomorphic in
,
such that
.
If the functions
can be selected (in some neighbourhood
)
so that the rank of the Jacobi matrix
at the point
is
,
is called a
regular point
of the analytic set
;
the number
is called the (complex)
dimension of
at
and is denoted by
.
The set
of all regular points of an analytic set
is an open everywhere-dense subset of
(in the induced topology on
as a subset of
).
Its complement
— the set of
singular points
of
— is an
analytic set in
that is nowhere dense in
.
By definition
the
dimension
of the analytic set
is the number
An analytic set
is called
pure
-dimensional
if
for all
.
For each
,
the set
is a pure
-dimensional
analytic set in
.
Thus, any analytic set in
can be represented as a finite union of
pure analytic sets,
.
At the singular points
,
so that the dimension of the analytic set of singular points of a pure
-dimensional
analytic set in
is smaller than
.
The connected components of
are complex manifolds. Since this is also true for the analytic set
,
one obtains the decomposition:
of the analytic set into complex manifolds. The decomposition
is more convenient (the dimensions of the summands strictly diminish,
);
it is called the
stratification
of
;
the connected components of the
-th
summand of this sum are called the
-dimensional
strata
of the analytic set
.
An analytic set
is called
reducible
(in
)
if it is the union of two analytic sets in
other than itself; otherwise it is called
irreducible
(in
).
All irreducible analytic sets in
are connected and pure. An analytic set
in
is irreducible if and only if the set
of its regular points is connected. The closure of each connected component of
is an irreducible analytic set in
;
such analytic sets are called the
irreducible components
of
.
All analytic sets in
are locally finite unions of their irreducible components.
If two analytic sets have no common irreducible
components, the dimension of their intersection is strictly smaller than
the dimension of each set. If the intersection of two irreducible sets in
contains a set that is open in each one, these analytic sets are identical (the
identity theorem).
An analytic set
in
is called
irreducible at a point
if there exists a fundamental system of neighbourhoods
of the point
in
such that all analytic sets
in
are irreducible;
is then called an
irreducibility point
of the analytic set
.
In a neighbourhood of each irreducibility point the analytic set has the structure of an
analytic cover,
i.e. for each such point
,
,
there exist a connected neighbourhood
,
a linear mapping
and an analytic set
such that the restriction of
to
is a proper mapping into
,
while the restriction of
to
is a finite-to-one locally biholomorphic cover over
.
For irreducible one-dimensional analytic sets there exists
thus (after a suitable linear change in coordinates)
a local parametric representation of the form
where
,
is a positive integer and the functions
are holomorphic in the disc
.
Thus, in a neighbourhood of each irreducibility point, a one-dimensional analytic
set is a topological manifold. For an analytic set
of higher dimension this is usually not true.
The union of a finite number and the intersection of any family of analytic sets in
are analytic sets in
.
Any analytic set in
is closed in
.
Any compact analytic set in
consists of a finite number of points. If
is connected and the analytic set
,
then
is open, everywhere-dense in
and is also connected. The set of all isolated points of an analytic set in
has no limit points in
.
Moreover, all analytic sets are locally connected.
A connected analytic set is pathwise connected.
Any analytic set in
of dimension
has locally in
finite
-dimensional
Hausdorff measure
.
If
,
there exist positive constants
and
(which depend on
and
)
such that
for all sufficiently small
.
The family of analytic sets is invariant under biholomorphic mappings (cf.
Biholomorphic mapping).
Moreover, if
is an analytic set in
and if
is a
proper holomorphic mapping,
then
is an analytic set in
.
The definition of analytic sets on complex manifolds is similar to the definition for
,
and all the properties listed above are preserved except
for one: in the general case there exist compact
non-discrete analytic sets. In special manifolds analytic sets can
have certain additional properties. For example, in the complex
-dimensional
projective space all analytic sets are algebraic, i.e. coincide with the
set of common zeros of some finite tuple of homogeneous polynomials
(Chow's theorem).
Real-analytic sets in open subsets of
are defined in the same manner, except that real-analytic functions must be
taken instead of holomorphic functions. Any real-analytic set is the
intersection of some analytic set (in some open subset of
)
with the real subspace
.