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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" />.
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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
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<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>
+
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 [[#References|[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|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  $  t $
 +
with the critical values lead to (strongly or weakly) distinguished bases of the vanishing homology group  $  H _ {n- 1 }  ( X _ {t} ) $.
  
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  $  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
  
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]]].
+
$$
 +
i ^ {- 1 } K  \rightarrow  R \Psi _ {f} K  \rightarrow  R \Phi _ {f} K  \mathop \rightarrow \limits ^ { {+ 1 }}
 +
$$
 +
 
 +
in the derived category  $  D _ {c}  ^ {b} ( X _ {0} ) $.
 +
Here  $  R \Phi _ {f} $
 +
is the vanishing cycle functor associated to  $  f $[[#References|[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 [[#References|[a5]]], [[#References|[a6]]] (see also [[D-module| $  D $-
 +
module]]; [[Derived category|Derived category]]). They play a crucial role in the theory of mixed Hodge modules [[#References|[a7]]].
  
 
====References====
 
====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>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S. Lefschetz, "L'analysis situs et la géométrie algébrique" , Gauthier-Villars (1924) {{MR|0033557}} {{MR|1520618}} {{ZBL|}} </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) {{MR|966191}} {{ZBL|0659.58002}} </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) {{MR|0923114}} {{ZBL|0683.32001}} </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) {{MR|0354657}} {{ZBL|}} </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) {{MR|0409883}} {{ZBL|}} </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) {{MR|0907933}} {{ZBL|}} </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 {{MR|1047741}} {{MR|1047415}} {{ZBL|0727.14004}} {{ZBL|0726.14007}} </TD></TR></table>

Latest revision as of 08:27, 6 June 2020


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
How to Cite This Entry:
Vanishing cycle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vanishing_cycle&oldid=13059
This article was adapted from an original article by J. Steenbrink (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article