Vanishing cycle

Let $ X $ be an $ n $- dimensional complex manifold with boundary, $ U $ a Riemann surface and $ f : X \rightarrow U $ a proper holomorphic mapping which has no critical points on the boundary of $ X $ and only non-degenerate critical points on the interior with distinct critical values. Let $ \gamma $ be a path on $ U $ such that $ \gamma ( 0) $ is a critical value of $ f $ but $ \gamma ( \tau ) $ is a regular value for $ \tau \in ( 0, 1] $. For $ V\subset [ 0, 1] $, write $ X _ {V} = \{ {( x, \tau ) \in X \times V } : {f( x) = \gamma ( \tau ) } \} $. The group $ H _ {n} ( X _ {[ 0,1] } , X _ {1} ) $ is then infinite cyclic. An $ n $- chain $ \Delta $ on $ X _ {[ 0,1] } $ generating it is called a Lefschetz thimble and its boundary $ \delta = \partial \Delta \in H _ {n- 1 } ( X _ {1} ) $ a (Lefschetz) vanishing cycle [a1]. It is uniquely determined by $ \gamma $ 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 $ t $ with the critical values lead to (strongly or weakly) distinguished bases of the vanishing homology group $ H _ {n- 1 } ( X _ {t} ) $.

If $ f : X \rightarrow \mathbf C $ is a holomorphic function on a complex space $ X $ and $ K $ is a constructible sheaf complex on $ X $, one obtains a constructible sheaf complex $ R \Psi _ {f} ( K) $ on $ X _ {0} = f ^ { - 1 } ( 0) $ in the following way. Let $ H \rightarrow \mathbf C ^ {*} $ be a universal covering and let $ k : X \times _ {\mathbf C ^ {*} } H \rightarrow X $, $ i : X _ {0} \rightarrow X $ be the natural mappings. Then $ R \Psi _ {f} K = i ^ {- 1 } Rk _ {*} k ^ {- 1 } K $. The functor $ R \Psi _ {f} $ is called the nearby cycle functor. There is a distinguished triangle

$$ i ^ {- 1 } K \rightarrow R \Psi _ {f} K \rightarrow R \Phi _ {f} K \mathop \rightarrow \limits ^ { {+ 1 }} $$

in the derived category $ D _ {c} ^ {b} ( X _ {0} ) $. Here $ R \Phi _ {f} $ is the vanishing cycle functor associated to $ f $[a4].

If the sheaf complex $ K $ is perverse, the same holds for $ R \Psi _ {f} K $ and $ R \Phi _ {f} K $. If $ X $ is a complex manifold, by the Riemann–Hilbert correspondence one has vanishing and nearby cycle functors $ \phi _ {f} $ and $ \psi _ {f} $ in the category of regular holonomic $ D _ {X} $- modules [a5], [a6] (see also $ D $- 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 -modules cohérents" , Hermann (1989) MR0907933
[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