View source for Vanishing cycle

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v0960701.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v0960702.png" />-dimensional complex manifold with boundary, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v0960703.png" /> a Riemann surface and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v0960704.png" /> a proper holomorphic mapping which has no critical points on the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v0960705.png" /> and only non-degenerate critical points on the interior with distinct critical values. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v0960706.png" /> be a path on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v0960707.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v0960708.png" /> is a critical value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v0960709.png" /> but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v09607010.png" /> is a regular value for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v09607011.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v09607012.png" />, write <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v09607013.png" />. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v09607014.png" /> is then infinite cyclic. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v09607015.png" />-chain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v09607016.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v09607017.png" /> generating it is called a Lefschetz thimble and its boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v09607018.png" /> a (Lefschetz) vanishing cycle [[#References|[a1]]]. It is uniquely determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v09607019.png" /> 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|Monodromy transformation]]) and of semi-universal deformations of isolated complete intersection singularities [[#References|[a2]]], [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v09607020.png" /> with the critical values lead to (strongly or weakly) distinguished bases of the vanishing homology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v09607021.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v09607022.png" /> is a holomorphic function on a complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v09607023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v09607024.png" /> is a constructible sheaf complex on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v09607025.png" />, one obtains a constructible sheaf complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v09607026.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v09607027.png" /> in the following way. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v09607028.png" /> be a universal covering and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v09607029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v09607030.png" /> be the natural mappings. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v09607031.png" />. The functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v09607032.png" /> is called the nearby cycle functor. There is a distinguished triangle

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v09607033.png" /></td> </tr></table>

in the derived category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v09607034.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v09607035.png" /> is the vanishing cycle functor associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v09607036.png" /> [[#References|[a4]]].

If the sheaf complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v09607037.png" /> is perverse, the same holds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v09607038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v09607039.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v09607040.png" /> is a complex manifold, by the Riemann–Hilbert correspondence one has vanishing and nearby cycle functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v09607041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v09607042.png" /> in the category of regular holonomic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v09607043.png" />-modules [[#References|[a5]]], [[#References|[a6]]] (see also [[D-module|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v09607044.png" />-module]]; [[Derived category|Derived category]]). They play a crucial role in the theory of mixed Hodge modules [[#References|[a7]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Lefschetz,   "L'analysis situs et la géométrie algébrique" , Gauthier-Villars  (1924)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Ebeling,   "The monodromy groups of isolated singularities of complete intersections" , ''Lect. notes in math.'' , '''1293''' , Springer  (1987)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P. Deligne (ed.)  N.M. Katz (ed.) , ''Groupes de monodromie en géométrie algébrique. SGA 7.II'' , ''Lect. notes in math.'' , '''340''' , Springer  (1973)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  B. Malgrange,   "Le polynôme de I.N. Bernstein d'une singularité isolée" , ''Lect. notes in math.'' , '''459''' , Springer  (1976)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  Z. Mebkhout,   "Systèmes différentiels. Le formalisme des six opérations de Grothendieck pour les <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096070/v09607045.png" />-modules cohérents" , Hermann  (1989)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  M. Saito,   "Mixed Hodge modules"  ''Publ. R.I.M.S. Kyoto Univ.'' , '''26'''  (1990)  pp. 221–333</TD></TR></table>