Exceptional analytic set

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 and also 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). For $ A $ to be exceptional it is sufficient that $ N $ be weakly negative (cf. 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. [1], [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 $[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 $ 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. [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. [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 [5].

References

[1]	H. Grauert, "Ueber Modifikationen und exzeptionelle analytische Mengen" Math. Ann. , 146 (1962) pp. 331–368 Zbl 0178.42702 Zbl 0173.33004
[2]	V. Ancona, "Un teorema di contrattibilità relativa" Boll. Unione Mat. Ital. , 9 : 3 (1974) pp. 785–790 MR0374483 Zbl 0327.32002
[3]	A. Fujiki, S. Nakano, "Supplement to "On the inverse of monodial transformations" " Publ. Res. Inst. Math. Sci. , 7 : 3 (1972) pp. 637–644 MR0294712 Zbl 0234.32019
[4]	A. Fujiki, "On the blowing down of analytic spaces" Publ. Res. Inst. Math. Sci. , 10 : 2 (1975) pp. 473–507 MR0374484 Zbl 0316.32009
[5]	K. Takijima, T. Suzuki, "On the trivial extension of equivalence relations on analytic spaces" Trans. Amer. Math. Soc. , 219 (1976) pp. 369–377 MR0412463 Zbl 0342.32006
[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 MR0374485 Zbl 0336.32011

References

[a1]	R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001