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 , 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 –).
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 , , ). 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 ,
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 ).
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Monodromy transformation. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Monodromy_transformation&oldid=24509