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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]]]).
 
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]]]).
  
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" />.
+
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 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 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
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====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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J. Milnor,   "Singular points of complex hypersurfaces" , Princeton Univ. Press (1968)</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)</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</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</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> J. Steenbrink,   "Limits of Hodge structures" ''Invent. Math.'' , '''31''' (1976) pp. 229–257</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</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> S. Lefschetz,   "L'analysis situs et la géométrie algébrique" , Gauthier-Villars (1924)</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</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 18:59, 15 April 2012

A transformation of the fibres (or of their homotopy invariants) of a fibre space corresponding to a path in the base. More precisely, let be a locally trivial fibre space and let be a path in with initial point and end-point . A trivialization of the fibration defines a homeomorphism of the fibre onto the fibre , . If the trivialization of is modified, then changes into a homotopically-equivalent homeomorphism; this also happens if is changed to a homotopic path. The homotopy type of is called the monodromy transformation corresponding to a path . When , that is, when is a loop, the monodromy transformation is a homeomorphism of 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 , is also called a monodromy transformation. The correspondence of with gives a representation of the fundamental group on .

The idea of a monodromy transformation arose in the study of multi-valued functions (see Monodromy theorem). If is the Riemann surface of such a function, then by eliminating the singular points of the function from the Riemann sphere , 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 be the unit disc in the complex plane, let be an analytic space, let be a proper holomorphic mapping (cf. Proper morphism), let be the fibre , , , and let . Diminishing, if necessary, the radius of , the fibre space can be made locally trivial. The monodromy transformation associated with a circuit around in is called the monodromy of the family at , it acts on the (co)homology spaces of the fibre , where . The most studied case is when and the fibres , , are smooth. The action of on , in this case, is quasi-unipotent [4], that is, there are positive integers and such that . The properties of the monodromy display many characteristic features of the degeneracy of the family . The monodromy of the family is closely related to mixed Hodge structures (cf. Hodge structure) on the cohomology spaces and (see [5][7]).

When the singularities of are isolated, the monodromy transformation can be localized. Let be a singular point of (or, equivalently, of ) and let be a sphere of sufficiently small radius in with centre at . Diminishing, if necessary, the radius of , a local trivialization of the fibre space can be defined. It is compatible with the trivialization of the fibre space on the boundary. This gives a diffeomorphism of the manifold of "vanishing cycles" into itself which is the identity on , and which is called the local monodromy of at . The action of the monodromy transformation on the cohomology spaces reflects the singularity of at (see [1], [2], [7]). It is known that the manifold is homotopically equivalent to a bouquet of -dimensional spheres, where and is the Milnor number of the germ of at .

The simplest case is that of a Morse singularity when, in a neighbourhood of , reduces to the form (cf. Morse lemma). In this case , and the interior of is diffeomorphic to the tangent bundle of the -dimensional sphere . A vanishing cycle is a generator of the cohomology group with compact support , defined up to sign. In general, if is a proper holomorphic mapping (as above, having a unique Morse singularity at ), then a cycle vanishing at is the image of a cycle under the natural mapping . In this case the specialization homomorphism is an isomorphism for , and the sequence

is exact. The monodromy transformation acts trivially on for and its action on is given by the Picard–Lefschetz formula: For ,

The sign in this formula and the values of are collected in the table.'
<tbody> </tbody>

A monodromy transformation preserves the intersection form on .

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 be a smooth manifold of dimension , and let , , be a pencil of hyperplane sections of with basic set (axis of the pencil) ; let the following conditions be satisfied: a) is a smooth submanifold in ; b) there is a finite set such that is smooth for ; and c) for the manifold has a unique non-degenerate quadratic singular point , where . Pencils with these properties (Lefschetz pencils) always exist. Let be a monoidal transformation with centre on the axis of the pencil, and let be the morphism defined by the pencil ; here for all . Let a point be fixed; then the monodromy transformation gives an action of on (non-trivial only for ). To describe the action of the monodromy on one chooses points , situated near , and paths leading from to . Let be the loop constructed as follows: first go along , then once round and, finally, return along to . In addition, let be a cycle vanishing at (more precisely, take a vanishing cycle in and transfer it to by means of the monodromy transformation corresponding to the path ). Finally, let be the subspace generated by the vanishing cycles , (the vanishing cohomology space). Then the following hold.

1) is generated by the elements , ;

2) the action of is given by the formula

3) the space is invariant under the action of the monodromy group ;

4) the space of elements in that are invariant relative to monodromy coincides with the orthogonal complement of relative to the intersection form on , and also with the images of the natural homomorphisms and ;

5) the vanishing cycles are conjugate (up to sign) under the action of ;

6) the action of on is absolutely irreducible.

The formalism of vanishing cycles, monodromy transformations and the Picard–Lefschetz theory has also been constructed for -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://www.encyclopediaofmath.org/index.php?title=Monodromy_transformation&oldid=24509
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article