Monodromy transformation

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

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[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
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