3. the monodromy

Let $f \in {\mathbb{C}}\{x_1, \dots, x_n\}$ be a convergent power series (in practice a polynomial) with isolated singularity at 0 and $\mu = \dim_{\mathbb{C}}{\mathbb{C}}\{x\}/\langle f_{x_1}, \dots, f_{x_n}\rangle$ the Milnor number of $f$.

Then $f$ defines in an $\varepsilon$-ball $B_\varepsilon$ around 0 a holomorphic function to ${\mathbb{C}}$, $f : B_\varepsilon \longrightarrow {\mathbb{C}}$.

The simple, counterclockwise path $\gamma$ in ${\mathbb{C}}$ around 0 induces a $C^\infty$-diffeomorphism of $X_t\; (t \not= 0)$ (as indicated in the figure) and an automorphism of the singular cohomology group $H^n(X_t, {\mathbb{C}})$ which is, by a theorem of Milnor, a $\mu$-dimensional ${\mathbb{C}}$-vector space. This automorphism

$$T : H^n(X_t, {\mathbb{C}}) \stackrel{\cong}{\longrightarrow } H^n(X_t, {\mathbb{C}})$$

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

The first important theorem is the

Monodromy theorem (Deligne 1970, Brieskorn 1971): The eigenvalues of $T$ are roots of unity, that is, we have

$$T = e^{2\pi i M},$$

where $M$ is a complex matrix with eigenvalues in ${\mathbb{Q}}$.

Hence, we are left with the problem of computing the Jordan normal form of $M$.

It is not at all clear that the purely topological definition of $T$ allows an algebraic and computable interpretation. The first hint in this direction is that we can compute $\dim_{\mathbb{C}}H^n(X_t, {\mathbb{C}})$, according to Milnor's theorem, algebraically by the formula for $\mu$ given above.

Since $X_t$ 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

$$H' = \Omega^n/df \wedge \Omega^{n-1} + d \Omega^{n-1},$$

$$H^{\prime\prime} = \Omega^{n+1}/df \wedge d \Omega^{n-1}$$

are free ${\mathbb{C}}\{t\}$-modules (via $f^\ast : {\mathbb{C}}\{t\} \longrightarrow {\mathbb{C}}\{f\} \subset {\mathbb{C}}\{x\}$) of rank equal to $\mu$. Here $(\Omega^\bullet,d)$ denotes the complex of holomorphic differential forms in $({\mathbb{C}}^n,0)$. $H^\prime$ and $H^{\prime\prime}$ are called Brieskorn lattices.

We define the local Gauß-Manin connection of $f$ as

$$\bigtriangledown : df \wedge H^\prime = df \wedge \Omega^n/df \wedge d \Omega^{n-1} \longrightarrow H^{\prime\prime},$$

$$\bigtriangledown [df \wedge \omega] = [d \omega].$$

Note that $\bigtriangledown(df \wedge H^\prime) \not\subset df \wedge H^\prime$, that is, $\bigtriangledown$ has a pole at 0. Tensoring with ${\mathbb{C}}\,(t)$, the quotient field of ${\mathbb{C}}\{t\}$, we can extend $\bigtriangledown$ to a meromorphic connection

$$\bigtriangledown : H^{\prime\prime} \underset{{\mathbb{C}}\{t\}}{... ...tarrow H^{\prime\prime} \underset{{\mathbb{C}}\{t\}}{\otimes} {\mathbb{C}}(t)$$

(since df $\wedge H^\prime \otimes {\mathbb{C}}(t) = H^{\prime\prime}\otimes {\mathbb{C}}(t)$) using the Leibnitz rule $\bigtriangledown (\omega y) = \bigtriangledown (\omega) y + \omega dy/dt$.

With respect to a basis $\omega_1, \dots, \omega_\mu$ of $H^{\prime\prime}$ we have $\bigtriangledown (\omega_i) = \underset{j}{\sum} a_{j i} \omega_j$ and, for any $\omega = \underset{i}{\sum} \omega_i y_i$, $\bigtriangledown (\omega) = \underset{i,j}{\sum} a_{ji} y_i + \underset{i}{\sum} \omega_i dy_i/dt$. Hence, the kernel of $\bigtriangledown$, together with a basis of $H^{\prime\prime}$, is the same as the solutions of the system of rank $\mu$ of ordinary differential equations

$$\dfrac{dy}{dt} = -Ay,\qquad A = (a_{ij}) \in$$    Mat$$\bigl(\mu \times \mu, {\mathbb{C}}(t)\bigr)$$

in a neighbourhood of 0 in ${\mathbb{C}}$. The connection matrix, $A = \underset{i \ge -p}{\sum} A_i t^i$, $A_i \in$   Mat$\,(\mu \times \mu, {\mathbb{C}})$, has a pole at $t = 0$ and is holomorphic for $t\not= 0$. If $\phi_t = (\phi_1, \dots, \phi_\mu)$ is a fundamental system of solutions at a point $t\not= 0$, then the analytic continuation of $\phi_t$ along the path $\gamma$ transforms $\phi_t$ into another fundamental system $\phi'_t$ which satisfies $\phi'_t = T_\bigtriangledown \phi_t$ for some matrix $T_\bigtriangledown \in$    GL$(\mu, {\mathbb{C}})$.

Fundamental fact (Brieskorn, 1971): The Picard-Lefschetz monodromy $T$ coincides with the monodromy $T_\bigtriangledown$ 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 $T$. Results of Gerard and Levelt allowed the extension of this algorithm to compute the Jordan normal form of $T$, 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 $H^{\prime\prime} \otimes {\mathbb{C}}(t)$ such that the connection matrix $A$ has pole of order 1.

Basically, if $A = A_{-1} t^{-1} + A_0 + A_1 t + \cdots$ has a simple pole, then $T = e^{2\pi i} A_{-1}$ is the monodromy (this holds if the eigenvalues of $A_{-1}$ 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 $K[x]_{\langle x\rangle}$ up to sufficiently high order;
4. computation of the connection matrix on increasing lattices in $H^{\prime\prime} \otimes {\mathbb{C}}(t)$ up to sufficiently high order (until saturation) by linear algebra over ${\mathbb{Q}}$;
5. computation of the transformation matrix to a simple pole by linear algebra over ${\mathbb{Q}}$;
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.