Difference between revisions of "Monodromy 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]]]). | 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 | + | 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"> | + | <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 & 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> |
Revision as of 16:59, 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 |
Monodromy transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Monodromy_transformation&oldid=18530