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3. the monodromy

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 to , .

The simple, counterclockwise path in around 0 induces a -diffeomorphism of (as indicated in the figure) and an automorphism of the singular cohomology group which is, by a theorem of Milnor, a -dimensional -vector space. This automorphism

is called the local Picard-Lefschetz monodromy of . We address the problem of computing the Jordan normal form of .

The first important theorem is the

Monodromy theorem (Deligne 1970, Brieskorn 1971): 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 Jordan normal form of .

It is not at all clear that the purely topological definition of allows an algebraic and computable interpretation. The first hint in this direction is that we can compute , according to Milnor's theorem, algebraically by the formula for given above.

Since is a complex Stein manifold, its complex cohomology can be computed, via the holomorphic de Rham theorem, with holomorphic differential forms, which is the starting point for computing the monodromy.

To cut a long story short, we just mention, cf. Br,Gre1 that

are free -modules (via ) of rank equal to . Here denotes the complex of holomorphic differential forms in . and are called Brieskorn lattices.

We define the local Gauß-Manin connection of as

Note that , that is, has a pole at 0. Tensoring with , the quotient field of , we can extend to a meromorphic connection

(since df ) using the Leibnitz rule .

With respect to a basis of we have and, for any , . Hence, the kernel of , together with a basis of , is the same as the solutions of the system of rank of ordinary differential equations

Mat

in a neighbourhood of 0 in . The connection matrix, ,    Mat, has a pole at and is holomorphic for . If is a fundamental system of solutions at a point , then the analytic continuation of along the path transforms into another fundamental system which satisfies for some matrix     GL.

Fundamental fact (Brieskorn, 1971): The Picard-Lefschetz monodromy coincides with the monodromy of the Gauß-Manin connection.

Brieskorn used this fact to describe in Br the essential steps for an algorithm to compute the characteristic polynomial of . Results of Gerard and Levelt allowed the extension of this algorithm to compute the Jordan normal form of , cf. GL. An early implementation by Nacken in Maple was not very efficient. Recently, Schu implemented an improved version in SINGULAR which is able to compute interesting examples.

The algorithm uses another basic theorem, the

Regularity Theorem (Brieskorn, 1971): The Gauß-Manin connection has a regular singular point at 0, that is, there exists a basis of some lattice in such that the connection matrix has 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:


LIB "mondromy.lib";
ring R   = 0,(x,y),ds;
poly f   = x2y2+x6+y6;   //example of A'Campo (monodromy is not diagonalisable)
matrix M = monodromy(f);
print(jordanform(M));
==> 1/2,1,  0,  0,  0,  0,  0,0,0,0,  0,  0,  0,
==> 0,  1/2,0,  0,  0,  0,  0,0,0,0,  0,  0,  0,
==> 0,  0,  2/3,0,  0,  0,  0,0,0,0,  0,  0,  0,
==> 0,  0,  0,  2/3,0,  0,  0,0,0,0,  0,  0,  0,
==> 0,  0,  0,  0,  5/6,0,  0,0,0,0,  0,  0,  0,
==> 0,  0,  0,  0,  0,  5/6,0,0,0,0,  0,  0,  0,
==> 0,  0,  0,  0,  0,  0,  1,0,0,0,  0,  0,  0,
==> 0,  0,  0,  0,  0,  0,  0,1,0,0,  0,  0,  0,
==> 0,  0,  0,  0,  0,  0,  0,0,1,0,  0,  0,  0,
==> 0,  0,  0,  0,  0,  0,  0,0,0,7/6,0,  0,  0,
==> 0,  0,  0,  0,  0,  0,  0,0,0,0,  7/6,0,  0,
==> 0,  0,  0,  0,  0,  0,  0,0,0,0,  0,  4/3,0,
==> 0,  0,  0,  0,  0,  0,  0,0,0,0,  0,  0,  4/3

Ingredients for the monodromy algorithm:

1. Computation of standard bases and normal forms for local orderings;
2. computation of Milnor number;
3. Taylor expansion of units in up to sufficiently high order;
4. computation of the connection matrix on increasing lattices in up to sufficiently high order (until saturation) by linear algebra over ;
5. computation of the transformation matrix to a simple pole by linear algebra over ;
6. factorisation of univariate polynomials (for Jordan normal form).

The most expensive parts are certain normal form computations for a local ordering and the linear algebra part because here one has to deal iteratively with matrices with several thousand rows and columns. It turned out that the SINGULAR implementation of modules (considered as sparse matrices) and the Buchberger inter-reduction is sufficiently efficient (though not the best possible) for such tasks.

Next: 7. Computer algebra solutions Up: 6. Some local algorithms Previous: 2. deformations   Contents
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