Exceptional analytic set
An analytic set in a complex space
for which there exists an analytic mapping
such that
is a point in the complex space
, while
is an analytic isomorphism. The modification
is called a contraction of
to
.
The problem of characterizing exceptional sets arose in algebraic geometry in relation to the study of birational transformations (cf. Birational transformation and also Exceptional subvariety). Very general criteria for exceptional sets have been found in analytic geometry. More precisely, let be a connected compact analytic set of positive dimension in a complex space
. The set
is exceptional if and only if there is a relatively-compact pseudo-convex neighbourhood of it in
in which it is a maximal compact analytic subset.
Let be a coherent sheaf of ideals whose zero set coincides with
and let
be the restriction to
of the linear space over
dual to
(cf. Vector bundle, analytic). For
to be exceptional it is sufficient that
be weakly negative (cf. Positive vector bundle). If
is a manifold and
is a submanifold of it, then
is the normal bundle over
. Sometimes, the bundle
being weakly negative is also necessary (e.g. if
is a submanifold of codimension 1, isomorphic to
, or if
is a two-dimensional manifold). In particular, a curve
on a complex surface
is exceptional if and only if the intersection matrix
of its irreducible components is negative definite (cf. [1], [2]). The structure of a neighbourhood of an exceptional analytic set
is completely determined by the ringed space
for sufficiently large
. Exceptional analytic sets have the following transitiveness condition: If
is a compact analytic space in
and is exceptional in
, while
is exceptional in
, then
is exceptional in
[6]. There are relative generalizations of the concept of an exceptional analytic set. These consider, roughly speaking, the simultaneous contraction of a family of analytic sets in an analytic family of complex spaces. An analogue of Grauert's criterion mentioned above is valid in this case (cf. [2]).
Another natural generalization of the concept of an exceptional analytic set is as follows. Let be a subspace in
and let a proper surjective holomorphic mapping
be given. A contraction of
along
is a proper surjective holomorphic mapping
, where
contains
as a subspace, such that
and
induces an isomorphism
. If
is a manifold of dimension
,
is a compact submanifold of codimension one in it, and
is a fibration with fibre
,
, then a necessary and sufficient condition for
to be contractible along
onto a manifold
is: The normal bundle
over
(which in this case coincides with the bundle corresponding to the divisor
) must induce a bundle
on each fibre
, where
is determined by a hyperplane in
. The corresponding contraction is the inverse to the monoidal transformation with centre at
(cf. [3]). On the other hand, for each modification
, where
is a manifold,
is a submanifold of it,
, and
is an isomorphism, the mapping
is a fibration with fibre
. Criteria for contractibility along
, as well as in more general situations, are known (cf. [4]). If
is exceptional in
and is a holomorphic retract of it (e.g.
is the zero section of a weakly-negative vector bundle), then
has a contraction along any
. If, moreover, the dimensions of the fibres of the retract
are equal to at least
, one can completely recover the initial space from the data
obtained after contraction [5].
References
[1] | H. Grauert, "Ueber Modifikationen und exzeptionelle analytische Mengen" Math. Ann. , 146 (1962) pp. 331–368 |
[2] | V. Ancona, "Un teorema di contrattibilità relativa" Boll. Unione Mat. Ital. , 9 : 3 (1974) pp. 785–790 |
[3] | A. Fujiki, S. Nakano, "Supplement to "On the inverse of monodial transformations" " Publ. Res. Inst. Math. Sci. , 7 : 3 (1972) pp. 637–644 |
[4] | A. Fujiki, "On the blowing down of analytic spaces" Publ. Res. Inst. Math. Sci. , 10 : 2 (1975) pp. 473–507 |
[5] | K. Takijima, T. Suzuki, "On the trivial extension of equivalence relations on analytic spaces" Trans. Amer. Math. Soc. , 219 (1976) pp. 369–377 |
[6] | V.A. Krasnov, "Transitivity of exceptional subspaces" Math. USSR-Izv. , 9 : 1 (1975) pp. 13–20 Izv. Akad. Nauk SSSR Ser. Mat. , 39 : 1 (1975) pp. 15–22 |
Comments
References
[a1] | R. Hartshorne, "Algebraic geometry" , Springer (1977) |
Exceptional analytic set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Exceptional_analytic_set&oldid=15887