Vanishing cycle
Let be an
-dimensional complex manifold with boundary,
a Riemann surface and
a proper holomorphic mapping which has no critical points on the boundary of
and only non-degenerate critical points on the interior with distinct critical values. Let
be a path on
such that
is a critical value of
but
is a regular value for
. For
, write
. The group
is then infinite cyclic. An
-chain
on
generating it is called a Lefschetz thimble and its boundary
a (Lefschetz) vanishing cycle [a1]. It is uniquely determined by
up to sign. Two cases are of particular importance: the case of a Lefschetz pencil of hyperplane sections of a projective variety (see Monodromy transformation) and of semi-universal deformations of isolated complete intersection singularities [a2], [a3]. In the latter case, one first restricts the semi-universal deformation to a smooth curve which intersects the discriminant transversely. Suitable choices of paths connecting a regular value
with the critical values lead to (strongly or weakly) distinguished bases of the vanishing homology group
.
If is a holomorphic function on a complex space
and
is a constructible sheaf complex on
, one obtains a constructible sheaf complex
on
in the following way. Let
be a universal covering and let
,
be the natural mappings. Then
. The functor
is called the nearby cycle functor. There is a distinguished triangle
![]() |
in the derived category . Here
is the vanishing cycle functor associated to
[a4].
If the sheaf complex is perverse, the same holds for
and
. If
is a complex manifold, by the Riemann–Hilbert correspondence one has vanishing and nearby cycle functors
and
in the category of regular holonomic
-modules [a5], [a6] (see also
-module; Derived category). They play a crucial role in the theory of mixed Hodge modules [a7].
References
[a1] | S. Lefschetz, "L'analysis situs et la géométrie algébrique" , Gauthier-Villars (1924) MR0033557 MR1520618 |
[a2] | V.I. Arnol'd, S.M. [S.M. Khusein-Zade] Gusein-Zade, A.N. Varchenko, "Singularities of differentiable maps" , 2 , Birkhäuser (1988) (Translated from Russian) MR966191 Zbl 0659.58002 |
[a3] | W. Ebeling, "The monodromy groups of isolated singularities of complete intersections" , Lect. notes in math. , 1293 , Springer (1987) MR0923114 Zbl 0683.32001 |
[a4] | P. Deligne (ed.) N.M. Katz (ed.) , Groupes de monodromie en géométrie algébrique. SGA 7.II , Lect. notes in math. , 340 , Springer (1973) MR0354657 |
[a5] | B. Malgrange, "Le polynôme de I.N. Bernstein d'une singularité isolée" , Lect. notes in math. , 459 , Springer (1976) MR0409883 |
[a6] | Z. Mebkhout, "Systèmes différentiels. Le formalisme des six opérations de Grothendieck pour les ![]() |
[a7] | M. Saito, "Mixed Hodge modules" Publ. R.I.M.S. Kyoto Univ. , 26 (1990) pp. 221–333 MR1047741 MR1047415 Zbl 0727.14004 Zbl 0726.14007 |
Vanishing cycle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vanishing_cycle&oldid=49109