The monodromy of a morphism between complex spaces or algebraic schemes/, which we suppose to be a differentiable fibre bundle outside the discriminant , describes the action of the fundamental group of on the cohomology of the general fibre. The Gauß-Manin connection may be considered as an algebraic description of the monodromy action by means of differential forms. Finally, the mixed Hodge structure is an analytic structure on generalising the Hodge decomposition of compact, smooth algebraic varieties. These concepts have many applications and were widely studied in the global situation for proper maps as well as in the local situation for isolated singularities, for a survey see [Ku]. Here we shall only consider the local case.
Let be a convergent power series (in practice a polynomial) with isolated singularity at 0 and the Milnor number of . Then defines in an -ball around 0 a holomorphic function , and, by a theorem of Milnor, there exists a small -disc in around such that is a -fibre bundle so that the general fibre , , is homotopy equivalent to a bouquet of -dimensional spheres.
The simple, counterclockwise path in around induces a -diffeomorphism of () and an automorphism of the singular cohomology group which is a -dimensional -vector space. The automorphism is called the local Picard-Lefschetz monodromy of . We address the problem of computing the eigenvalues and the Jordan normal form of .
The first important theorem is the monodromy theorem, due to Deligne in the global and to Brieskorn in the local situation which says that the eigenvalues of are roots of unity, that is, we have , where is a complex matrix with eigenvalues in .
Hence, we are left with the problem of computing the eigenvalues and the
Jordan normal form of .
Since is a complex Stein manifold, its complex cohomology can be
computed, via the holomorphic de Rham theorem, by using holomorphic
forms, which is the starting point of Brieskorn's algorithm for computing the
monodromy. To cut a long story short, we just mention that the
Brieskorn lattices (cf. [Br])
It is a fundamental fact that the Picard-Lefschetz monodromy coincides with the monodromy of the Gauß-Manin connection.
Brieskorn [Br] used this fact to describe the essential steps for an algorithm to compute the characteristic polynomial of . Results of Gerard and Levelt [GL] allowed the extension of this algorithm to compute the Jordan normal form of . An implementation of Schulze in SINGULAR is able to compute interesting examples (including the uni- and bimodal singularities, [Sch]).
The algorithm uses the regularity theorem which says that there exists a basis of some lattice in such that the connection matrix has a pole of order 1.
Basically, if has a simple pole, then is the monodromy (this holds if the eigenvalues of do not differ by integers which can be achieved algorithmically).
SINGULAR example for computing the monodromy (omitting the output):
> LIB "mondromy.lib"; > ring R = 0,(x,y),ds; > poly f = x2y2+x6+y6; //example of A'Campo (monodromy is not > matrix M = monodromy(f); //diagonalisable) > print(jordanform(M)); //prints Jordan normalform of monodromyIngredients for the implementation of Brieskorn's algorithm:
In the remaining part of this section we describe a new algorithm, developed by
M. Schulze, based on the theory of -modules (
the complex with differential defined by
By definition of the differential , computing up to order amounts to expressing times an element of in the basis and , which is the Jacobian ideal of . This can be done using Gröbner basis methods. To do the -th step in the computation of the saturation of , one has to compute up to order . To compute the residue of on , whose eigenvalues are the eigenvalues of monodromy, one has to compute up to sufficiently high order and compute a -basis of as well as the basis representation of the images of this basis under with respect to this basis. This can also be done using Gröbner basis methods.
Compared to the Brieskorn algorithm, we have interchanged the roles of and . The -structure of is much more natural and there are many advantages of this new algorithm: There are no problems with estimations, no huge linear algebra problems, we need not to lift a power of in the Jacobian ideal, the basis of is easier to compute, and so on. The main point is that we can continue the computation when we have to increase the order of . In the Brieskorn algorithm, we have to start again almost from the beginning. Nevertheless, the three components of this new algorithm explained above also require difficult computations, especially the first one. The new algorithm can be extended to compute the Jordan normal form of the monodromy in a similar way as it was done in [Sch].