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A transformation of the fibres (or of their homotopy invariants) of a fibre space corresponding to a path in the base. More precisely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m0647001.png" /> be a locally trivial [[Fibre space|fibre space]] and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m0647002.png" /> be a path in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m0647003.png" /> with initial point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m0647004.png" /> and end-point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m0647005.png" />. A trivialization of the fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m0647006.png" /> defines a homeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m0647007.png" /> of the fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m0647008.png" /> onto the fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m0647009.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470010.png" />. If the trivialization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470011.png" /> is modified, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470012.png" /> changes into a homotopically-equivalent homeomorphism; this also happens if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470013.png" /> is changed to a homotopic path. The homotopy type of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470014.png" /> is called the monodromy transformation corresponding to a path <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470015.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470016.png" />, that is, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470017.png" /> is a loop, the monodromy transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470018.png" /> is a homeomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470019.png" /> into itself (defined, yet again, up to a homotopy). This mapping, and also the homomorphisms induced by it on the homology and cohomology spaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470020.png" />, is also called a monodromy transformation. The correspondence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470021.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470022.png" /> gives a representation of the [[Fundamental group|fundamental group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470023.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470024.png" />.
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The idea of a monodromy transformation arose in the study of multi-valued functions (see [[Monodromy theorem|Monodromy theorem]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470025.png" /> is the [[Riemann surface|Riemann surface]] of such a function, then by eliminating the singular points of the function from the Riemann sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470026.png" />, an unbranched covering is obtained. The monodromy transformation in this case is also called a covering or deck transformation.
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The monodromy transformation arises most frequently in the following situation. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470027.png" /> be the unit disc in the complex plane, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470028.png" /> be an [[Analytic space|analytic space]], let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470029.png" /> be a proper holomorphic mapping (cf. [[Proper morphism|Proper morphism]]), let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470030.png" /> be the fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470033.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470034.png" />. Diminishing, if necessary, the radius of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470035.png" />, the fibre space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470036.png" /> can be made locally trivial. The monodromy transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470037.png" /> associated with a circuit around <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470038.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470039.png" /> is called the monodromy of the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470040.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470041.png" />, it acts on the (co)homology spaces of the fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470042.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470043.png" />. The most studied case is when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470044.png" /> and the fibres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470046.png" />, are smooth. The action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470047.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470048.png" />, in this case, is quasi-unipotent [[#References|[4]]], that is, there are positive integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470050.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470051.png" />. The properties of the monodromy display many characteristic features of the degeneracy of the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470052.png" />. The monodromy of the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470053.png" /> is closely related to mixed Hodge structures (cf. [[Hodge structure|Hodge structure]]) on the cohomology spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470055.png" /> (see [[#References|[5]]]–[[#References|[7]]]).
+
A transformation of the fibres (or of their homotopy invariants) of a fibre space corresponding to a path in the base. More precisely, let $  p : E \rightarrow B $
 +
be a locally trivial [[Fibre space|fibre space]] and let $  \gamma : [ 0 , 1 ] \rightarrow B $
 +
be a path in  $  B $
 +
with initial point  $  a = \gamma ( 0) $
 +
and end-point  $  b = \gamma ( 1) $.  
 +
A trivialization of the fibration  $  \gamma  ^ {*} ( E) $
 +
defines a homeomorphism  $  T _  \gamma  $
 +
of the fibre $  p  ^ {-} 1 ( a) $
 +
onto the fibre  $  p  ^ {-} 1 ( b) $,
 +
$  T _  \gamma  : p  ^ {-} 1 ( a) \rightarrow p  ^ {-} 1 ( b) $.  
 +
If the trivialization of $  \gamma  ^ {*} ( E) $
 +
is modified, then  $  T _  \gamma  $
 +
changes into a homotopically-equivalent homeomorphism; this also happens if  $  \gamma $
 +
is changed to a homotopic path. The homotopy type of  $  T _  \gamma  $
 +
is called the monodromy transformation corresponding to a path  $  \gamma $.  
 +
When  $  a = b $,  
 +
that is, when  $  \gamma $
 +
is a loop, the monodromy transformation  $  T _  \gamma  $
 +
is a homeomorphism of  $  F = p  ^ {-} 1 ( a) $
 +
into itself (defined, yet again, up to a homotopy). This mapping, and also the homomorphisms induced by it on the homology and cohomology spaces of $  F $,
 +
is also called a monodromy transformation. The correspondence of  $  \gamma $
 +
with  $  T _  \gamma  $
 +
gives a representation of the [[Fundamental group|fundamental group]] $  \pi _ {1} ( B , a ) $
 +
on $  H  ^ {*} ( ) $.
  
When the singularities of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470056.png" /> are isolated, the monodromy transformation can be localized. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470057.png" /> be a singular point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470058.png" /> (or, equivalently, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470059.png" />) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470060.png" /> be a sphere of sufficiently small radius in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470061.png" /> with centre at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470062.png" />. Diminishing, if necessary, the radius of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470063.png" />, a local trivialization of the fibre space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470064.png" /> can be defined. It is compatible with the trivialization of the fibre space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470065.png" /> on the boundary. This gives a diffeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470066.png" /> of the manifold of "vanishing cycles" <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470067.png" /> into itself which is the identity on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470068.png" />, and which is called the local monodromy of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470069.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470070.png" />. The action of the monodromy transformation on the cohomology spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470071.png" /> reflects the singularity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470072.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470073.png" /> (see [[#References|[1]]], [[#References|[2]]], [[#References|[7]]]). It is known that the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470074.png" /> is homotopically equivalent to a bouquet of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470075.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470076.png" />-dimensional spheres, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470077.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470078.png" /> is the Milnor number of the germ of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470079.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470080.png" />.
+
The idea of a monodromy transformation arose in the study of multi-valued functions (see [[Monodromy theorem|Monodromy theorem]]). If  $  S \rightarrow P  ^ {1} ( \mathbf C ) $
 +
is the [[Riemann surface|Riemann surface]] of such a function, then by eliminating the singular points of the function from the Riemann sphere  $  P  ^ {1} ( \mathbf C ) $,
 +
an unbranched covering is obtained. The monodromy transformation in this case is also called a covering or deck transformation.
  
The simplest case is that of a Morse singularity when, in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470081.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470082.png" /> reduces to the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470083.png" /> (cf. [[Morse lemma|Morse lemma]]). In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470084.png" />, and the interior <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470085.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470086.png" /> is diffeomorphic to the tangent bundle of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470087.png" />-dimensional sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470088.png" />. A vanishing cycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470089.png" /> is a generator of the cohomology group with compact support <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470090.png" />, defined up to sign. In general, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470091.png" /> is a proper holomorphic mapping (as above, having a unique Morse singularity at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470092.png" />), then a cycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470093.png" /> vanishing at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470094.png" /> is the image of a cycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470095.png" /> under the natural mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470096.png" />. In this case the specialization homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470097.png" /> is an isomorphism for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m06470098.png" />, and the sequence
+
The monodromy transformation arises most frequently in the following situation. Let  $  D = \{ {z \in \mathbf C } : {| z | < 1 } \} $
 +
be the unit disc in the complex plane, let  $  X $
 +
be an [[Analytic space|analytic space]], let  $  f : X \rightarrow D $
 +
be a proper holomorphic mapping (cf. [[Proper morphism|Proper morphism]]), let  $  X _ {t} $
 +
be the fibre  $  f ^ { - 1 } ( t) $,
 +
$  t \in D $,
 +
$  D  ^ {*} = D \setminus  \{ 0 \} $,
 +
and let  $  X  ^ {*} = f ^ { - 1 } ( D  ^ {*} ) $.  
 +
Diminishing, if necessary, the radius of  $  D $,  
 +
the fibre space  $  f : X  ^ {*} \rightarrow D  ^ {*} $
 +
can be made locally trivial. The monodromy transformation  $  T $
 +
associated with a circuit around  $  0 $
 +
in  $  D $
 +
is called the monodromy of the family  $  f : X \rightarrow D $
 +
at  $  0 \in D $,
 +
it acts on the (co)homology spaces of the fibre  $  X _ {t} $,  
 +
where  $  t \in D  ^ {*} $.  
 +
The most studied case is when  $  X $
 +
and the fibres  $  X _ {t} $,  
 +
$  t \neq 0 $,
 +
are smooth. The action of  $  T $
 +
on  $  H  ^ {*} ( X _ {t} , \mathbf Q ) $,
 +
in this case, is quasi-unipotent [[#References|[4]]], that is, there are positive integers  $  k $
 +
and  $  N $
 +
such that  $  ( T  ^ {k} - 1 )  ^ {N} = 0 $.  
 +
The properties of the monodromy display many characteristic features of the degeneracy of the family  $  f : X \rightarrow D $.  
 +
The monodromy of the family  $  f : X \rightarrow D $
 +
is closely related to mixed Hodge structures (cf. [[Hodge structure|Hodge structure]]) on the cohomology spaces  $  H  ^ {*} ( X _ {0} ) $
 +
and  $  H  ^ {*} ( X _ {t} ) $(
 +
see [[#References|[5]]]–[[#References|[7]]]).
  
<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/m/m064/m064700/m06470099.png" /></td> </tr></table>
+
When the singularities of  $  f : X \rightarrow D $
 +
are isolated, the monodromy transformation can be localized. Let  $  x $
 +
be a singular point of  $  f $(
 +
or, equivalently, of  $  X _ {0} $)
 +
and let  $  B $
 +
be a sphere of sufficiently small radius in  $  X $
 +
with centre at  $  x $.
 +
Diminishing, if necessary, the radius of  $  D $,
 +
a local trivialization of the fibre space  $  B \cap f ^ { - 1 } ( D  ^ {*} ) $
 +
can be defined. It is compatible with the trivialization of the fibre space  $  \partial  B \cap f ^ { - 1 } ( D) \rightarrow D $
 +
on the boundary. This gives a diffeomorphism  $  T $
 +
of the manifold of "vanishing cycles" $  V _ {t} = B \cap X _ {t} $
 +
into itself which is the identity on  $  \partial  V _ {t} $,
 +
and which is called the local monodromy of  $  f $
 +
at  $  x $.  
 +
The action of the monodromy transformation on the cohomology spaces  $  H  ^ {*} ( V _ {t} ) $
 +
reflects the singularity of  $  f $
 +
at  $  x $(
 +
see [[#References|[1]]], [[#References|[2]]], [[#References|[7]]]). It is known that the manifold  $  V _ {t} $
 +
is homotopically equivalent to a bouquet of  $  \mu $
 +
$  n $-
 +
dimensional spheres, where  $  n + 1 =  \mathop{\rm dim}  X $
 +
and  $  \mu $
 +
is the Milnor number of the germ of  $  f $
 +
at  $  x $.
  
<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/m/m064/m064700/m064700100.png" /></td> </tr></table>
+
The simplest case is that of a Morse singularity when, in a neighbourhood of  $  x $,
 +
$  f $
 +
reduces to the form  $  f = z _ {0}  ^ {2} + \dots + z _ {n}  ^ {2} $(
 +
cf. [[Morse lemma|Morse lemma]]). In this case  $  \mu = 1 $,
 +
and the interior  $  V _ {t}  ^ {0} $
 +
of  $  V _ {t} $
 +
is diffeomorphic to the tangent bundle of the  $  n $-
 +
dimensional sphere  $  S  ^ {n} $.
 +
A vanishing cycle  $  \delta $
 +
is a generator of the cohomology group with compact support  $  H _ {c}  ^ {n} ( V _ {t}  ^ {0} , \mathbf Z ) \cong \mathbf Z $,
 +
defined up to sign. In general, if  $  f : X \rightarrow D $
 +
is a proper holomorphic mapping (as above, having a unique Morse singularity at  $  x $),
 +
then a cycle  $  \delta _ {x} $
 +
vanishing at  $  x $
 +
is the image of a cycle  $  \delta \in H _ {c}  ^ {n} ( V _ {t}  ^ {0} ) $
 +
under the natural mapping  $  H _ {c}  ^ {n} ( V _ {t}  ^ {0} ) \rightarrow H  ^ {n} ( X _ {t} ) $.
 +
In this case the specialization homomorphism  $  r _ {t}  ^ {*} : H  ^ {i} ( X _ {0} ) \rightarrow H  ^ {i} ( X _ {t} ) $
 +
is an isomorphism for  $  i \neq n , n + 1 $,
 +
and the sequence
  
is exact. The monodromy transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700101.png" /> acts trivially on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700102.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700103.png" /> and its action on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700104.png" /> is given by the Picard–Lefschetz formula: For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700105.png" />,
+
$$
 +
0 \rightarrow  H  ^ {n} ( X _ {0} )  \rightarrow  H  ^ {n} ( X _ {t} )
 +
  \mathop \rightarrow \limits ^ { {(  , \delta _ {x} ) }}  \mathbf Z \rightarrow
 +
$$
  
<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/m/m064/m064700/m064700106.png" /></td> </tr></table>
+
$$
 +
\rightarrow \
 +
H  ^ {n+} 1 ( X _ {0} )  \rightarrow  H  ^ {n+} 1 ( X _ {t} )  \rightarrow  0
 +
$$
  
The sign in this formula and the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700107.png" /> are collected in the table.''''''<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700108.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700109.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700110.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700111.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700112.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700113.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700114.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700115.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700116.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700117.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700118.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700119.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700120.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700121.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700122.png" /></td> </tr> </tbody> </table>
+
is exact. The monodromy transformation  $  T $
 +
acts trivially on  $  H  ^ {i} ( X _ {t} ) $
 +
for  $  i \neq n $
 +
and its action on  $  H  ^ {n} ( X _ {t} ) $
 +
is given by the Picard–Lefschetz formula: For  $  z \in H  ^ {n} ( X _ {t} ) $,
 +
 
 +
$$
 +
T _ {z}  =  z \pm  ( z , \delta _ {x} ) \delta _ {x} .
 +
$$
 +
 
 +
The sign in this formula and the values of $  ( \delta _ {x} , \delta _ {x} ) $
 +
are collected in the table.<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  n  \mathop{\rm mod}  4 $
 +
</td> <td colname="2" style="background-color:white;" colspan="1"> 0 $
 +
</td> <td colname="3" style="background-color:white;" colspan="1"> $  1 $
 +
</td> <td colname="4" style="background-color:white;" colspan="1"> $  2 $
 +
</td> <td colname="5" style="background-color:white;" colspan="1"> $  3 $
 +
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  \pm  $
 +
</td> <td colname="2" style="background-color:white;" colspan="1"> $  - $
 +
</td> <td colname="3" style="background-color:white;" colspan="1"> $  - $
 +
</td> <td colname="4" style="background-color:white;" colspan="1"> $  + $
 +
</td> <td colname="5" style="background-color:white;" colspan="1"> $  + $
 +
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  ( \delta _ {x} , \delta _ {x} ) $
 +
</td> <td colname="2" style="background-color:white;" colspan="1"> $  2 $
 +
</td> <td colname="3" style="background-color:white;" colspan="1"> 0 $
 +
</td> <td colname="4" style="background-color:white;" colspan="1"> $  - 2 $
 +
</td> <td colname="5" style="background-color:white;" colspan="1"> 0 $
 +
</td> </tr> </tbody> </table>
  
 
</td></tr> </table>
 
</td></tr> </table>
  
A monodromy transformation preserves the intersection form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700123.png" />.
+
A monodromy transformation preserves the intersection form on $  H  ^ {n} ( X _ {t} ) $.
  
Vanishing cycles and monodromy transformations are used in the Picard–Lefschetz theory, associating the cohomology space of a projective complex manifold and its hyperplane sections. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700124.png" /> be a smooth manifold of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700125.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700126.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700127.png" />, be a pencil of hyperplane sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700128.png" /> with basic set (axis of the pencil) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700129.png" />; let the following conditions be satisfied: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700130.png" /> is a smooth submanifold in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700131.png" />; b) there is a finite set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700132.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700133.png" /> is smooth for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700134.png" />; and c) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700135.png" /> the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700136.png" /> has a unique non-degenerate quadratic singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700137.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700138.png" />. Pencils with these properties (Lefschetz pencils) always exist. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700139.png" /> be a monoidal transformation with centre on the axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700140.png" /> of the pencil, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700141.png" /> be the morphism defined by the pencil <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700142.png" />; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700143.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700144.png" />. Let a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700145.png" /> be fixed; then the monodromy transformation gives an action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700146.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700147.png" /> (non-trivial only for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700148.png" />). To describe the action of the monodromy on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700149.png" /> one chooses points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700150.png" />, situated near <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700151.png" />, and paths <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700152.png" /> leading from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700153.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700154.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700155.png" /> be the loop constructed as follows: first go along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700156.png" />, then once round <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700157.png" /> and, finally, return along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700158.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700159.png" />. In addition, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700160.png" /> be a cycle vanishing at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700161.png" /> (more precisely, take a vanishing cycle in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700162.png" /> and transfer it to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700163.png" /> by means of the monodromy transformation corresponding to the path <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700164.png" />). Finally, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700165.png" /> be the subspace generated by the vanishing cycles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700166.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700167.png" /> (the vanishing cohomology space). Then the following hold.
+
Vanishing cycles and monodromy transformations are used in the Picard–Lefschetz theory, associating the cohomology space of a projective complex manifold and its hyperplane sections. Let $  X \subset  P  ^ {N} $
 +
be a smooth manifold of dimension $  n + 1 $,  
 +
and let $  \{ X _ {t} \} $,  
 +
$  t \in P  ^ {1} $,  
 +
be a pencil of hyperplane sections of $  X $
 +
with basic set (axis of the pencil) $  Y \subset  X $;  
 +
let the following conditions be satisfied: a) $  Y $
 +
is a smooth submanifold in $  X $;  
 +
b) there is a finite set $  S \subset  P  ^ {1} $
 +
such that $  X _ {t} $
 +
is smooth for $  t \in P  ^ {1} \setminus  S $;  
 +
and c) for $  s \in S $
 +
the manifold $  X _ {s} $
 +
has a unique non-degenerate quadratic singular point $  x _ {s} $,  
 +
where $  x _ {s} \in Y $.  
 +
Pencils with these properties (Lefschetz pencils) always exist. Let $  \sigma : \overline{X}\; \rightarrow X $
 +
be a monoidal transformation with centre on the axis $  Y $
 +
of the pencil, and let $  f : \overline{X}\; \rightarrow P  ^ {1} $
 +
be the morphism defined by the pencil $  \{ X _ {t} \} $;  
 +
here $  f ^ { - 1 } ( t) \cong X _ {t} $
 +
for all $  t \in P  ^ {1} $.  
 +
Let a point 0 \in P  ^ {1} \setminus  S $
 +
be fixed; then the monodromy transformation gives an action of $  \pi _ {1} ( P  ^ {1} \setminus  S , 0 ) $
 +
on $  H  ^ {i} ( X _ {0} ) $(
 +
non-trivial only for $  i = n $).  
 +
To describe the action of the monodromy on $  H  ^ {n} ( X _ {0} ) $
 +
one chooses points $  s  ^  \prime  $,  
 +
situated near $  s \in S $,  
 +
and paths $  \gamma _ {s}  ^  \prime  $
 +
leading from 0 $
 +
to $  s  ^  \prime  $.  
 +
Let $  \gamma _ {s} \in \pi _ {1} ( p  ^ {1} \setminus  S , 0 ) $
 +
be the loop constructed as follows: first go along $  \gamma _ {s}  ^  \prime  $,  
 +
then once round $  s $
 +
and, finally, return along $  \gamma _ {s}  ^  \prime  $
 +
to 0 $.  
 +
In addition, let $  \delta _ {s} $
 +
be a cycle vanishing at $  x _ {s} $(
 +
more precisely, take a vanishing cycle in $  H  ^ {n} ( X _ {s  ^  \prime  } ) $
 +
and transfer it to $  H  ^ {n} ( X _ {0} ) $
 +
by means of the monodromy transformation corresponding to the path $  \gamma _ {s}  ^  \prime  $).  
 +
Finally, let $  E \subset  H  ^ {n} ( X _ {0} , \mathbf Q ) $
 +
be the subspace generated by the vanishing cycles $  \delta _ {s} $,  
 +
$  s \in S $(
 +
the vanishing cohomology space). Then the following hold.
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700168.png" /> is generated by the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700169.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700170.png" />;
+
1) $  \pi _ {1} ( P  ^ {1} \setminus  S , 0 ) $
 +
is generated by the elements $  \gamma _ {s} $,  
 +
$  s \in S $;
  
2) the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700171.png" /> is given by the formula
+
2) the action of $  \gamma _ {s} $
 +
is given by the formula
  
<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/m/m064/m064700/m064700172.png" /></td> </tr></table>
+
$$
 +
\gamma _ {s} ( z )  = z \pm  ( z , \delta _ {s} ) \delta _ {s} ;
 +
$$
  
3) the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700173.png" /> is invariant under the action of the monodromy group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700174.png" />;
+
3) the space $  E \subset  H  ^ {n} ( X _ {0} ) $
 +
is invariant under the action of the monodromy group $  \pi _ {1} ( P  ^ {1} \setminus  S , 0 ) $;
  
4) the space of elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700175.png" /> that are invariant relative to monodromy coincides with the orthogonal complement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700176.png" /> relative to the intersection form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700177.png" />, and also with the images of the natural homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700178.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700179.png" />;
+
4) the space of elements in $  H  ^ {n} ( X _ {0} ) $
 +
that are invariant relative to monodromy coincides with the orthogonal complement of $  E $
 +
relative to the intersection form on $  H  ^ {n} ( X _ {0} ) $,  
 +
and also with the images of the natural homomorphisms $  H _ {n} ( \overline{X}\; ) \rightarrow H _ {n} ( X _ {0} ) $
 +
and $  H  ^ {n} ( X ) \rightarrow H  ^ {n} ( X _ {0} ) $;
  
5) the vanishing cycles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700180.png" /> are conjugate (up to sign) under the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700181.png" />;
+
5) the vanishing cycles $  \pm  \delta _ {s} $
 +
are conjugate (up to sign) under the action of $  \pi _ {1} ( P  ^ {1} \setminus  S , 0 ) $;
  
6) the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700182.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700183.png" /> is absolutely irreducible.
+
6) the action of $  \pi _ {1} ( P  ^ {1} \setminus  S , 0 ) $
 +
on $  E $
 +
is absolutely irreducible.
  
The formalism of vanishing cycles, monodromy transformations and the Picard–Lefschetz theory has also been constructed for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064700/m064700184.png" />-adic cohomology spaces of algebraic varieties over any field (see [[#References|[3]]]).
+
The formalism of vanishing cycles, monodromy transformations and the Picard–Lefschetz theory has also been constructed for $  l $-
 +
adic cohomology spaces of algebraic varieties over any field (see [[#References|[3]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> V.I. Arnol'd, "Normal forms of functions in neighbourhoods of degenerate critical points" ''Russian Math. Surveys'' , '''29''' : 2 (1974) pp. 10–50 ''Uspekhi Mat. Nauk'' , '''29''' : 2 (1974) pp. 11–49 {{MR|}} {{ZBL|0304.57018}} {{ZBL|0298.57022}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J. Milnor, "Singular points of complex hypersurfaces" , Princeton Univ. Press (1968) {{MR|0239612}} {{ZBL|0184.48405}} </TD></TR><TR><TD valign="top">[3]</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">[4]</TD> <TD valign="top"> C.H. Clemens, "Picard–Lefschetz theorem for families of nonsingular algebraic varieties acquiring ordinary singularities" ''Trans. Amer. Math. Soc.'' , '''136''' (1969) pp. 93–108 {{MR|0233814}} {{ZBL|0185.51302}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> W. Schmid, "Variation of Hodge structure: the singularities of the period mapping" ''Invent. Math.'' , '''22''' (1973) pp. 211–319 {{MR|0382272}} {{ZBL|0278.14003}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> J. Steenbrink, "Limits of Hodge structures" ''Invent. Math.'' , '''31''' (1976) pp. 229–257 {{MR|0429885}} {{ZBL|0303.14002}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> J.H.M. Steenbrink, "Mixed Hodge structure on the vanishing cohomology" P. Holm (ed.) , ''Real and Complex Singularities (Oslo, 1976). Proc. Nordic Summer School'' , Sijthoff &amp; Noordhoff (1977) pp. 524–563 {{MR|0485870}} {{ZBL|0373.14007}} </TD></TR><TR><TD valign="top">[8]</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">[9]</TD> <TD valign="top"> S. Lefschetz, "A page of mathematical autobiography" ''Bull. Amer. Math. Soc.'' , '''74''' : 5 (1968) pp. 854–879 {{MR|0240803}} {{ZBL|0187.18601}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> V.I. Arnol'd, "Normal forms of functions in neighbourhoods of degenerate critical points" ''Russian Math. Surveys'' , '''29''' : 2 (1974) pp. 10–50 ''Uspekhi Mat. Nauk'' , '''29''' : 2 (1974) pp. 11–49 {{MR|}} {{ZBL|0304.57018}} {{ZBL|0298.57022}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J. Milnor, "Singular points of complex hypersurfaces" , Princeton Univ. Press (1968) {{MR|0239612}} {{ZBL|0184.48405}} </TD></TR><TR><TD valign="top">[3]</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">[4]</TD> <TD valign="top"> C.H. Clemens, "Picard–Lefschetz theorem for families of nonsingular algebraic varieties acquiring ordinary singularities" ''Trans. Amer. Math. Soc.'' , '''136''' (1969) pp. 93–108 {{MR|0233814}} {{ZBL|0185.51302}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> W. Schmid, "Variation of Hodge structure: the singularities of the period mapping" ''Invent. Math.'' , '''22''' (1973) pp. 211–319 {{MR|0382272}} {{ZBL|0278.14003}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> J. Steenbrink, "Limits of Hodge structures" ''Invent. Math.'' , '''31''' (1976) pp. 229–257 {{MR|0429885}} {{ZBL|0303.14002}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> J.H.M. Steenbrink, "Mixed Hodge structure on the vanishing cohomology" P. Holm (ed.) , ''Real and Complex Singularities (Oslo, 1976). Proc. Nordic Summer School'' , Sijthoff &amp; Noordhoff (1977) pp. 524–563 {{MR|0485870}} {{ZBL|0373.14007}} </TD></TR><TR><TD valign="top">[8]</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">[9]</TD> <TD valign="top"> S. Lefschetz, "A page of mathematical autobiography" ''Bull. Amer. Math. Soc.'' , '''74''' : 5 (1968) pp. 854–879 {{MR|0240803}} {{ZBL|0187.18601}} </TD></TR></table>

Latest revision as of 08:01, 6 June 2020


A transformation of the fibres (or of their homotopy invariants) of a fibre space corresponding to a path in the base. More precisely, let $ p : E \rightarrow B $ be a locally trivial fibre space and let $ \gamma : [ 0 , 1 ] \rightarrow B $ be a path in $ B $ with initial point $ a = \gamma ( 0) $ and end-point $ b = \gamma ( 1) $. A trivialization of the fibration $ \gamma ^ {*} ( E) $ defines a homeomorphism $ T _ \gamma $ of the fibre $ p ^ {-} 1 ( a) $ onto the fibre $ p ^ {-} 1 ( b) $, $ T _ \gamma : p ^ {-} 1 ( a) \rightarrow p ^ {-} 1 ( b) $. If the trivialization of $ \gamma ^ {*} ( E) $ is modified, then $ T _ \gamma $ changes into a homotopically-equivalent homeomorphism; this also happens if $ \gamma $ is changed to a homotopic path. The homotopy type of $ T _ \gamma $ is called the monodromy transformation corresponding to a path $ \gamma $. When $ a = b $, that is, when $ \gamma $ is a loop, the monodromy transformation $ T _ \gamma $ is a homeomorphism of $ F = p ^ {-} 1 ( a) $ into itself (defined, yet again, up to a homotopy). This mapping, and also the homomorphisms induced by it on the homology and cohomology spaces of $ F $, is also called a monodromy transformation. The correspondence of $ \gamma $ with $ T _ \gamma $ gives a representation of the fundamental group $ \pi _ {1} ( B , a ) $ on $ H ^ {*} ( F ) $.

The idea of a monodromy transformation arose in the study of multi-valued functions (see Monodromy theorem). If $ S \rightarrow P ^ {1} ( \mathbf C ) $ is the Riemann surface of such a function, then by eliminating the singular points of the function from the Riemann sphere $ P ^ {1} ( \mathbf C ) $, an unbranched covering is obtained. The monodromy transformation in this case is also called a covering or deck transformation.

The monodromy transformation arises most frequently in the following situation. Let $ D = \{ {z \in \mathbf C } : {| z | < 1 } \} $ be the unit disc in the complex plane, let $ X $ be an analytic space, let $ f : X \rightarrow D $ be a proper holomorphic mapping (cf. Proper morphism), let $ X _ {t} $ be the fibre $ f ^ { - 1 } ( t) $, $ t \in D $, $ D ^ {*} = D \setminus \{ 0 \} $, and let $ X ^ {*} = f ^ { - 1 } ( D ^ {*} ) $. Diminishing, if necessary, the radius of $ D $, the fibre space $ f : X ^ {*} \rightarrow D ^ {*} $ can be made locally trivial. The monodromy transformation $ T $ associated with a circuit around $ 0 $ in $ D $ is called the monodromy of the family $ f : X \rightarrow D $ at $ 0 \in D $, it acts on the (co)homology spaces of the fibre $ X _ {t} $, where $ t \in D ^ {*} $. The most studied case is when $ X $ and the fibres $ X _ {t} $, $ t \neq 0 $, are smooth. The action of $ T $ on $ H ^ {*} ( X _ {t} , \mathbf Q ) $, in this case, is quasi-unipotent [4], that is, there are positive integers $ k $ and $ N $ such that $ ( T ^ {k} - 1 ) ^ {N} = 0 $. The properties of the monodromy display many characteristic features of the degeneracy of the family $ f : X \rightarrow D $. The monodromy of the family $ f : X \rightarrow D $ is closely related to mixed Hodge structures (cf. Hodge structure) on the cohomology spaces $ H ^ {*} ( X _ {0} ) $ and $ H ^ {*} ( X _ {t} ) $( see [5][7]).

When the singularities of $ f : X \rightarrow D $ are isolated, the monodromy transformation can be localized. Let $ x $ be a singular point of $ f $( or, equivalently, of $ X _ {0} $) and let $ B $ be a sphere of sufficiently small radius in $ X $ with centre at $ x $. Diminishing, if necessary, the radius of $ D $, a local trivialization of the fibre space $ B \cap f ^ { - 1 } ( D ^ {*} ) $ can be defined. It is compatible with the trivialization of the fibre space $ \partial B \cap f ^ { - 1 } ( D) \rightarrow D $ on the boundary. This gives a diffeomorphism $ T $ of the manifold of "vanishing cycles" $ V _ {t} = B \cap X _ {t} $ into itself which is the identity on $ \partial V _ {t} $, and which is called the local monodromy of $ f $ at $ x $. The action of the monodromy transformation on the cohomology spaces $ H ^ {*} ( V _ {t} ) $ reflects the singularity of $ f $ at $ x $( see [1], [2], [7]). It is known that the manifold $ V _ {t} $ is homotopically equivalent to a bouquet of $ \mu $ $ n $- dimensional spheres, where $ n + 1 = \mathop{\rm dim} X $ and $ \mu $ is the Milnor number of the germ of $ f $ at $ x $.

The simplest case is that of a Morse singularity when, in a neighbourhood of $ x $, $ f $ reduces to the form $ f = z _ {0} ^ {2} + \dots + z _ {n} ^ {2} $( cf. Morse lemma). In this case $ \mu = 1 $, and the interior $ V _ {t} ^ {0} $ of $ V _ {t} $ is diffeomorphic to the tangent bundle of the $ n $- dimensional sphere $ S ^ {n} $. A vanishing cycle $ \delta $ is a generator of the cohomology group with compact support $ H _ {c} ^ {n} ( V _ {t} ^ {0} , \mathbf Z ) \cong \mathbf Z $, defined up to sign. In general, if $ f : X \rightarrow D $ is a proper holomorphic mapping (as above, having a unique Morse singularity at $ x $), then a cycle $ \delta _ {x} $ vanishing at $ x $ is the image of a cycle $ \delta \in H _ {c} ^ {n} ( V _ {t} ^ {0} ) $ under the natural mapping $ H _ {c} ^ {n} ( V _ {t} ^ {0} ) \rightarrow H ^ {n} ( X _ {t} ) $. In this case the specialization homomorphism $ r _ {t} ^ {*} : H ^ {i} ( X _ {0} ) \rightarrow H ^ {i} ( X _ {t} ) $ is an isomorphism for $ i \neq n , n + 1 $, and the sequence

$$ 0 \rightarrow H ^ {n} ( X _ {0} ) \rightarrow H ^ {n} ( X _ {t} ) \mathop \rightarrow \limits ^ { {( , \delta _ {x} ) }} \mathbf Z \rightarrow $$

$$ \rightarrow \ H ^ {n+} 1 ( X _ {0} ) \rightarrow H ^ {n+} 1 ( X _ {t} ) \rightarrow 0 $$

is exact. The monodromy transformation $ T $ acts trivially on $ H ^ {i} ( X _ {t} ) $ for $ i \neq n $ and its action on $ H ^ {n} ( X _ {t} ) $ is given by the Picard–Lefschetz formula: For $ z \in H ^ {n} ( X _ {t} ) $,

$$ T _ {z} = z \pm ( z , \delta _ {x} ) \delta _ {x} . $$

The sign in this formula and the values of $ ( \delta _ {x} , \delta _ {x} ) $

are collected in the table.

<tbody> </tbody>
$ n \mathop{\rm mod} 4 $ $ 0 $ $ 1 $ $ 2 $ $ 3 $
$ \pm $ $ - $ $ - $ $ + $ $ + $
$ ( \delta _ {x} , \delta _ {x} ) $ $ 2 $ $ 0 $ $ - 2 $ $ 0 $

A monodromy transformation preserves the intersection form on $ H ^ {n} ( X _ {t} ) $.

Vanishing cycles and monodromy transformations are used in the Picard–Lefschetz theory, associating the cohomology space of a projective complex manifold and its hyperplane sections. Let $ X \subset P ^ {N} $ be a smooth manifold of dimension $ n + 1 $, and let $ \{ X _ {t} \} $, $ t \in P ^ {1} $, be a pencil of hyperplane sections of $ X $ with basic set (axis of the pencil) $ Y \subset X $; let the following conditions be satisfied: a) $ Y $ is a smooth submanifold in $ X $; b) there is a finite set $ S \subset P ^ {1} $ such that $ X _ {t} $ is smooth for $ t \in P ^ {1} \setminus S $; and c) for $ s \in S $ the manifold $ X _ {s} $ has a unique non-degenerate quadratic singular point $ x _ {s} $, where $ x _ {s} \in Y $. Pencils with these properties (Lefschetz pencils) always exist. Let $ \sigma : \overline{X}\; \rightarrow X $ be a monoidal transformation with centre on the axis $ Y $ of the pencil, and let $ f : \overline{X}\; \rightarrow P ^ {1} $ be the morphism defined by the pencil $ \{ X _ {t} \} $; here $ f ^ { - 1 } ( t) \cong X _ {t} $ for all $ t \in P ^ {1} $. Let a point $ 0 \in P ^ {1} \setminus S $ be fixed; then the monodromy transformation gives an action of $ \pi _ {1} ( P ^ {1} \setminus S , 0 ) $ on $ H ^ {i} ( X _ {0} ) $( non-trivial only for $ i = n $). To describe the action of the monodromy on $ H ^ {n} ( X _ {0} ) $ one chooses points $ s ^ \prime $, situated near $ s \in S $, and paths $ \gamma _ {s} ^ \prime $ leading from $ 0 $ to $ s ^ \prime $. Let $ \gamma _ {s} \in \pi _ {1} ( p ^ {1} \setminus S , 0 ) $ be the loop constructed as follows: first go along $ \gamma _ {s} ^ \prime $, then once round $ s $ and, finally, return along $ \gamma _ {s} ^ \prime $ to $ 0 $. In addition, let $ \delta _ {s} $ be a cycle vanishing at $ x _ {s} $( more precisely, take a vanishing cycle in $ H ^ {n} ( X _ {s ^ \prime } ) $ and transfer it to $ H ^ {n} ( X _ {0} ) $ by means of the monodromy transformation corresponding to the path $ \gamma _ {s} ^ \prime $). Finally, let $ E \subset H ^ {n} ( X _ {0} , \mathbf Q ) $ be the subspace generated by the vanishing cycles $ \delta _ {s} $, $ s \in S $( the vanishing cohomology space). Then the following hold.

1) $ \pi _ {1} ( P ^ {1} \setminus S , 0 ) $ is generated by the elements $ \gamma _ {s} $, $ s \in S $;

2) the action of $ \gamma _ {s} $ is given by the formula

$$ \gamma _ {s} ( z ) = z \pm ( z , \delta _ {s} ) \delta _ {s} ; $$

3) the space $ E \subset H ^ {n} ( X _ {0} ) $ is invariant under the action of the monodromy group $ \pi _ {1} ( P ^ {1} \setminus S , 0 ) $;

4) the space of elements in $ H ^ {n} ( X _ {0} ) $ that are invariant relative to monodromy coincides with the orthogonal complement of $ E $ relative to the intersection form on $ H ^ {n} ( X _ {0} ) $, and also with the images of the natural homomorphisms $ H _ {n} ( \overline{X}\; ) \rightarrow H _ {n} ( X _ {0} ) $ and $ H ^ {n} ( X ) \rightarrow H ^ {n} ( X _ {0} ) $;

5) the vanishing cycles $ \pm \delta _ {s} $ are conjugate (up to sign) under the action of $ \pi _ {1} ( P ^ {1} \setminus S , 0 ) $;

6) the action of $ \pi _ {1} ( P ^ {1} \setminus S , 0 ) $ on $ E $ is absolutely irreducible.

The formalism of vanishing cycles, monodromy transformations and the Picard–Lefschetz theory has also been constructed for $ l $- adic cohomology spaces of algebraic varieties over any field (see [3]).

References

[1] V.I. Arnol'd, "Normal forms of functions in neighbourhoods of degenerate critical points" Russian Math. Surveys , 29 : 2 (1974) pp. 10–50 Uspekhi Mat. Nauk , 29 : 2 (1974) pp. 11–49 Zbl 0304.57018 Zbl 0298.57022
[2] J. Milnor, "Singular points of complex hypersurfaces" , Princeton Univ. Press (1968) MR0239612 Zbl 0184.48405
[3] P. Deligne (ed.) N.M. Katz (ed.) , Groupes de monodromie en géométrie algébrique. SGA 7.II , Lect. notes in math. , 340 , Springer (1973) MR0354657
[4] C.H. Clemens, "Picard–Lefschetz theorem for families of nonsingular algebraic varieties acquiring ordinary singularities" Trans. Amer. Math. Soc. , 136 (1969) pp. 93–108 MR0233814 Zbl 0185.51302
[5] W. Schmid, "Variation of Hodge structure: the singularities of the period mapping" Invent. Math. , 22 (1973) pp. 211–319 MR0382272 Zbl 0278.14003
[6] J. Steenbrink, "Limits of Hodge structures" Invent. Math. , 31 (1976) pp. 229–257 MR0429885 Zbl 0303.14002
[7] J.H.M. Steenbrink, "Mixed Hodge structure on the vanishing cohomology" P. Holm (ed.) , Real and Complex Singularities (Oslo, 1976). Proc. Nordic Summer School , Sijthoff & Noordhoff (1977) pp. 524–563 MR0485870 Zbl 0373.14007
[8] S. Lefschetz, "L'analysis situs et la géométrie algébrique" , Gauthier-Villars (1924) MR0033557 MR1520618
[9] S. Lefschetz, "A page of mathematical autobiography" Bull. Amer. Math. Soc. , 74 : 5 (1968) pp. 854–879 MR0240803 Zbl 0187.18601
How to Cite This Entry:
Monodromy transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Monodromy_transformation&oldid=47885
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article