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)
[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)
[a3]	W. Ebeling, "The monodromy groups of isolated singularities of complete intersections" , Lect. notes in math. , 1293 , Springer (1987)
[a4]	P. Deligne (ed.) N.M. Katz (ed.) , Groupes de monodromie en géométrie algébrique. SGA 7.II , Lect. notes in math. , 340 , Springer (1973)
[a5]	B. Malgrange, "Le polynôme de I.N. Bernstein d'une singularité isolée" , Lect. notes in math. , 459 , Springer (1976)
[a6]	Z. Mebkhout, "Systèmes différentiels. Le formalisme des six opérations de Grothendieck pour les -modules cohérents" , Hermann (1989)
[a7]	M. Saito, "Mixed Hodge modules" Publ. R.I.M.S. Kyoto Univ. , 26 (1990) pp. 221–333