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An analytic set  $  A $
in a complex space  $  X $
for which there exists an analytic mapping  $  f :  X \rightarrow Y $
such that  $  f ( A) = y $
is a point in the complex space  $  Y $, 
while  $  f :  X \setminus  A \rightarrow Y \setminus  \{ y \} $
is an analytic isomorphism. The modification  $  f $
is called a contraction of  $  A $
to  $  y $.

The problem of characterizing exceptional sets arose in algebraic geometry in relation to the study of birational transformations (cf. [[Birational transformation|Birational transformation]] and also [[Exceptional subvariety|Exceptional subvariety]]). Very general criteria for exceptional sets have been found in analytic geometry. More precisely, let  $  A $
be a connected compact analytic set of positive dimension in a complex space  $  X $. 
The set  $  A $
is exceptional if and only if there is a relatively-compact pseudo-convex neighbourhood of it in  $  X $
in which it is a maximal compact analytic subset.

Let  $  \mathfrak M $
be a coherent sheaf of ideals whose zero set coincides with  $  A $
and let  $  N $
be the restriction to  $  A $
of the linear space over  $  X $
dual to  $  \mathfrak M $(
cf. [[Vector bundle, analytic|Vector bundle, analytic]]). For  $  A $
to be exceptional it is sufficient that  $  N $
be weakly negative (cf. [[Positive vector bundle|Positive vector bundle]]). If  $  X $
is a manifold and  $  A $
is a submanifold of it, then  $  N $
is the normal bundle over  $  X $. 
Sometimes, the bundle  $  N $
being weakly negative is also necessary (e.g. if  $  A $
is a submanifold of codimension 1, isomorphic to  $  P  ^ {k} ( \mathbf C ) $, 
or if  $  X $
is a two-dimensional manifold). In particular, a curve  $  A $
on a complex surface  $  X $
is exceptional if and only if the intersection matrix  $  ( A _ {i} A _ {j} ) $
of its irreducible components is negative definite (cf. [[#References|[1]]], [[#References|[2]]]). The structure of a neighbourhood of an exceptional analytic set  $  A \subset  X $
is completely determined by the ringed space  $  ( A , {\mathcal O} _ {X} / \mathfrak M  ^  \mu  \mid  _ {A} ) $
for sufficiently large  $  \mu $. 
Exceptional analytic sets have the following transitiveness condition: If  $  B \subset  A $
is a compact analytic space in  $  X $
and is exceptional in  $  A $, 
while  $  A $
is exceptional in  $  X $, 
then  $  B $
is exceptional in  $  X $[[#References|[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. [[#References|[2]]]).

Another natural generalization of the concept of an exceptional analytic set is as follows. Let  $  A $
be a subspace in  $  X $
and let a proper surjective holomorphic mapping  $  \phi :  A \rightarrow B $
be given. A contraction of  $  X $
along  $  \phi $
is a proper surjective holomorphic mapping  $  f :  X \rightarrow Y $, 
where  $  Y $
contains  $  B $
as a subspace, such that  $  f \mid  _ {A} = \phi $
and  $  f $
induces an isomorphism  $  X \setminus  A \rightarrow Y \setminus  B $. 
If  $  X $
is a manifold of dimension  $  \geq  3 $, 
$  A $
is a compact submanifold of codimension one in it, and  $  \phi $
is a fibration with fibre  $  P  ^ {r} ( \mathbf C ) $, 
$  r > 1 $, 
then a necessary and sufficient condition for  $  X $
to be contractible along  $  \phi $
onto a manifold  $  Y $
is: The normal bundle  $  N $
over  $  A $(
which in this case coincides with the bundle corresponding to the divisor  $  A $) 
must induce a bundle  $  - L $
on each fibre  $  \phi  ^ {-} 1 ( b) \cong P  ^ {r} ( \mathbf C ) $, 
where  $  L $
is determined by a hyperplane in  $  P  ^ {r} ( \mathbf C ) $. 
The corresponding contraction is the inverse to the monoidal transformation with centre at  $  B $(
cf. [[#References|[3]]]). On the other hand, for each modification  $  f :  X \rightarrow Y $, 
where  $  Y $
is a manifold,  $  B = f ( A) $
is a submanifold of it,  $   \mathop{\rm dim}  B <  \mathop{\rm dim}  A $, 
and  $  f :  X \setminus  A \rightarrow X \setminus  B $
is an isomorphism, the mapping  $  f \mid  _ {A} $
is a fibration with fibre  $  P  ^ {r} ( \mathbf C ) $. 
Criteria for contractibility along  $  \phi $, 
as well as in more general situations, are known (cf. [[#References|[4]]]). If  $  A $
is exceptional in  $  X $
and is a holomorphic retract of it (e.g.  $  A $
is the zero section of a weakly-negative vector bundle), then  $  X $
has a contraction along any  $  \phi $. 
If, moreover, the dimensions of the fibres of the retract  $  X \rightarrow A $
are equal to at least  $   \mathop{\rm dim}  A + 2 $, 
one can completely recover the initial space from the data  $  Y $
obtained after contraction [[#References|[5]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Grauert, "Ueber Modifikationen und exzeptionelle analytische Mengen" ''Math. Ann.'' , '''146''' (1962) pp. 331–368 {{MR|}} {{ZBL|0178.42702}} {{ZBL|0173.33004}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> V. Ancona, "Un teorema di contrattibilità relativa" ''Boll. Unione Mat. Ital.'' , '''9''' : 3 (1974) pp. 785–790 {{MR|0374483}} {{ZBL|0327.32002}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A. Fujiki, S. Nakano, "Supplement to "On the inverse of monodial transformations" " ''Publ. Res. Inst. Math. Sci.'' , '''7''' : 3 (1972) pp. 637–644 {{MR|0294712}} {{ZBL|0234.32019}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A. Fujiki, "On the blowing down of analytic spaces" ''Publ. Res. Inst. Math. Sci.'' , '''10''' : 2 (1975) pp. 473–507 {{MR|0374484}} {{ZBL|0316.32009}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> K. Takijima, T. Suzuki, "On the trivial extension of equivalence relations on analytic spaces" ''Trans. Amer. Math. Soc.'' , '''219''' (1976) pp. 369–377 {{MR|0412463}} {{ZBL|0342.32006}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> 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 {{MR|0374485}} {{ZBL|0336.32011}} </TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>