View source for Exceptional analytic set

An analytic set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e0367701.png" /> in a complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e0367702.png" /> for which there exists an analytic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e0367703.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e0367704.png" /> is a point in the complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e0367705.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e0367706.png" /> is an analytic isomorphism. The modification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e0367707.png" /> is called a contraction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e0367708.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e0367709.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677010.png" /> be a connected compact analytic set of positive dimension in a complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677011.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677012.png" /> is exceptional if and only if there is a relatively-compact pseudo-convex neighbourhood of it in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677013.png" /> in which it is a maximal compact analytic subset.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677014.png" /> be a coherent sheaf of ideals whose zero set coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677015.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677016.png" /> be the restriction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677017.png" /> of the linear space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677018.png" /> dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677019.png" /> (cf. [[Vector bundle, analytic|Vector bundle, analytic]]). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677020.png" /> to be exceptional it is sufficient that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677021.png" /> be weakly negative (cf. [[Positive vector bundle|Positive vector bundle]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677022.png" /> is a manifold and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677023.png" /> is a submanifold of it, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677024.png" /> is the normal bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677025.png" />. Sometimes, the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677026.png" /> being weakly negative is also necessary (e.g. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677027.png" /> is a submanifold of codimension 1, isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677028.png" />, or if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677029.png" /> is a two-dimensional manifold). In particular, a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677030.png" /> on a complex surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677031.png" /> is exceptional if and only if the intersection matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677032.png" /> of its irreducible components is negative definite (cf. [[#References|[1]]], [[#References|[2]]]). The structure of a neighbourhood of an exceptional analytic set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677033.png" /> is completely determined by the ringed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677034.png" /> for sufficiently large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677035.png" />. Exceptional analytic sets have the following transitiveness condition: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677036.png" /> is a compact analytic space in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677037.png" /> and is exceptional in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677038.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677039.png" /> is exceptional in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677040.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677041.png" /> is exceptional in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677042.png" /> [[#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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677043.png" /> be a subspace in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677044.png" /> and let a proper surjective holomorphic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677045.png" /> be given. A contraction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677046.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677047.png" /> is a proper surjective holomorphic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677048.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677049.png" /> contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677050.png" /> as a subspace, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677052.png" /> induces an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677053.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677054.png" /> is a manifold of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677056.png" /> is a compact submanifold of codimension one in it, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677057.png" /> is a fibration with fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677059.png" />, then a necessary and sufficient condition for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677060.png" /> to be contractible along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677061.png" /> onto a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677062.png" /> is: The normal bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677063.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677064.png" /> (which in this case coincides with the bundle corresponding to the divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677065.png" />) must induce a bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677066.png" /> on each fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677067.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677068.png" /> is determined by a hyperplane in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677069.png" />. The corresponding contraction is the inverse to the monoidal transformation with centre at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677070.png" /> (cf. [[#References|[3]]]). On the other hand, for each modification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677071.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677072.png" /> is a manifold, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677073.png" /> is a submanifold of it, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677074.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677075.png" /> is an isomorphism, the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677076.png" /> is a fibration with fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677077.png" />. Criteria for contractibility along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677078.png" />, as well as in more general situations, are known (cf. [[#References|[4]]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677079.png" /> is exceptional in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677080.png" /> and is a holomorphic retract of it (e.g. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677081.png" /> is the zero section of a weakly-negative vector bundle), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677082.png" /> has a contraction along any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677083.png" />. If, moreover, the dimensions of the fibres of the retract <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677084.png" /> are equal to at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677085.png" />, one can completely recover the initial space from the data <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036770/e03677086.png" /> 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>