C.5 Gauss-Manin connection
==========================


Let $f\colon(C^{n+1},0)\rightarrow(C,0)$ be a complex isolated hypersurface singularity given by a polynomial with algebraic coefficients which we also denote by $f$.
Let $O=C[x_0,\ldots,x_n]_{(x_0,\ldots,x_n)}$ be the local ring at the origin and $J_f$ the Jacobian ideal of $f$.

A {\bf Milnor representative} of $f$ defines a differentiable fibre bundle over the punctured disc with fibres of homotopy type of $\mu$ $n$-spheres.
The $n$-th cohomology bundle is a flat vector bundle of dimension $n$ and carries a natural flat connection with covariant derivative $\partial_t$.
The {\bf monodromy operator} is the action of a positively oriented generator of the fundamental group of the puctured disc on the Milnor fibre.
Sections in the cohomology bundle of {\bf moderate growth} at $0$ form a regular $D=C\{t\}[\partial_t]$-module $G$, the {\bf Gauss-Manin connection}.

By integrating along flat multivalued families of cycles, one can consider fibrewise global holomorphic differential forms as elements of $G$.
This factors through an inclusion of the {\bf Brieskorn lattice} $H'':=\Omega^{n+1}_{C^{n+1},0}/df\wedge d\Omega^{n-1}_{C^{n+1},0}$ in $G$.

The $D$-module structure defines the {\bf V-filtration} $V$ on $G$ by $V^\alpha:=\sum_{\beta\ge\alpha}C\{t\}ker(t\partial_t-\beta)^{n+1}$.
The Brieskorn lattice defines the {\bf Hodge filtration} $F$ on $G$ by $F_k=\partial_t^kH''$ which comes from the {\bf mixed Hodge structure} on the Milnor fibre.
Note that $F_{-1}=H'$.

The induced V-filtration on the Brieskorn lattice determines the {\bf singularity spectrum} $Sp$ by $Sp(\alpha):=\dim_CGr_V^\alpha Gr^F_0G$.
The spectrum consists of $\mu$ rational numbers $\alpha_1,\dots,\alpha_\mu$ such that $e^{2\pi i\alpha_1},\dots,e^{2\pi i\alpha_\mu}$ are the eigenvalues of the monodromy.
These {\bf spectral numbers} lie in the open interval $(-1,n)$, symmetric about the midpoint $(n-1)/2$.

The spectrum is constant under $\mu$-constant deformations and has the following semicontinuity property:
The number of spectral numbers in an interval $(a,a+1]$ of all singularities of a small deformation of $f$ is greater or equal to that of f in this interval.
For semiquasihomogeneous singularities, this also holds for intervals of the form $(a,a+1)$.

Two given isolated singularities $f$ and $g$ determine two spectra and from these spectra we get an integer.
This integer is the maximal positive integer $k$ such that the semicontinuity holds for the spectrum of $f$ and $k$ times the spectrum of $g$.
These numbers give bounds for the maximal number of isolated singularities of a specific type on a hypersurface $X\subset{P}^n$ of degree $d$: 
such a hypersurface has a smooth hyperplane section, and the complement is a small deformation of a cone over this hyperplane section.
The cone itself being a $\mu$-constant deformation of $x_0^d+\dots+x_n^d=0$, the singularities are bounded by the spectrum of $x_0^d+\dots+x_n^d$.

Using the library {\tt gaussman.lib} one can compute the {\bf monodromy}, the V-filtration on $H''/H'$, and the spectrum.


Let us consider as an example 
$f=x^5+x^2y^2+y^5$
.
First, we compute a matrix 
$M$
 such that
$\exp(2\pi iM)$
is a monodromy matrix of 
$f$
 and the Jordan normal form of 
$M$
:
  LIB "gaussman.lib";
  ring R=0,(x,y),ds;
  poly f=x5+x2y2+y5;
  list l=monodromy(f);
  matrix M=jordanmatrix(l[1],l[2],l[3]);
  print(M);
==> 1/2,0,  0,   0,   0,   0,   0,0,    0,    0,    0,   
==> 1,  1/2,0,   0,   0,   0,   0,0,    0,    0,    0,   
==> 0,  0,  7/10,0,   0,   0,   0,0,    0,    0,    0,   
==> 0,  0,  0,   7/10,0,   0,   0,0,    0,    0,    0,   
==> 0,  0,  0,   0,   9/10,0,   0,0,    0,    0,    0,   
==> 0,  0,  0,   0,   0,   9/10,0,0,    0,    0,    0,   
==> 0,  0,  0,   0,   0,   0,   1,0,    0,    0,    0,   
==> 0,  0,  0,   0,   0,   0,   0,11/10,0,    0,    0,   
==> 0,  0,  0,   0,   0,   0,   0,0,    11/10,0,    0,   
==> 0,  0,  0,   0,   0,   0,   0,0,    0,    13/10,0,   
==> 0,  0,  0,   0,   0,   0,   0,0,    0,    0,    13/10

Now, we compute the V-filtration on 
$H''/H'$
 and the spectrum:
  LIB "gaussman.lib";
  ring R=0,(x,y),ds;
  poly f=x5+x2y2+y5;
  list l=vfilt(f);
  print(l[1]);
==> -1/2,
==> -3/10,
==> -1/10,
==> 0,
==> 1/10,
==> 3/10,
==> 1/2
  print(l[2]);
==> 1,2,2,1,2,2,1
  print(l[3]);
==> [1]:
==>    _[1]=gen(11)
==> [2]:
==>    _[1]=gen(10)
==>    _[2]=gen(6)
==> [3]:
==>    _[1]=gen(9)
==>    _[2]=gen(4)
==> [4]:
==>    _[1]=gen(5)
==> [5]:
==>    _[1]=gen(3)
==>    _[2]=gen(8)
==> [6]:
==>    _[1]=gen(2)
==>    _[2]=gen(7)
==> [7]:
==>    _[1]=gen(1)
  print(l[4]);
==> y5,
==> y4,
==> y3,
==> y2,
==> xy,
==> y,
==> x4,
==> x3,
==> x2,
==> x,
==> 1
Here l[1] contains the spectral numbers, l[2] the corresponding multiplicities, l[3] a 
$C$
-basis of the V-filtration on 
$H''/H'$
 in terms of the monomial basis of
$O/J_f\cong H''/H'$
in l[4].

If the principal part of $f$ is $C$-nondegenerate, one can compute the spectrum using the library {\tt spectrum.lib}.
In this case, the V-filtration on $H''$ coincides with the Newton-filtration on $H''$ which allows to compute the spectrum more efficiently.


Let us calculate one specific example, the maximal number 
of triple points of type
$\tilde{E}_6$ on a surface $X\subset{P}^3$
of degree seven.
This calculation can be done over the rationals.
So choose a local ordering on 
$Q[x,y,z]$
. Here we take the
negative degree lexicographical ordering which is denoted
ds in SINGULAR:

ring r=0,(x,y,z),ds;
LIB "spectrum.lib";
poly f=x^7+y^7+z^7;
list s1=spectrumnd( f );
s1;
==> [1]:
==>    _[1]=-4/7
==>    _[2]=-3/7
==>    _[3]=-2/7
==>    _[4]=-1/7
==>    _[5]=0
==>    _[6]=1/7
==>    _[7]=2/7
==>    _[8]=3/7
==>    _[9]=4/7
==>    _[10]=5/7
==>    _[11]=6/7
==>    _[12]=1
==>    _[13]=8/7
==>    _[14]=9/7
==>    _[15]=10/7
==>    _[16]=11/7
==> [2]:
==>    1,3,6,10,15,21,25,27,27,25,21,15,10,6,3,1

The command spectrumnd(f) computes the spectrum of 
$f$
 and
returns a list with six entries:
The Milnor number
$\mu(f)$, the geometric genus $p_g(f)$
and the number of different spectrum numbers.
The other three entries are of type intvec.
They contain the numerators, denominators and
multiplicities of the spectrum numbers. So
$x^7+y^7+z^7=0$
has Milnor number 216 and geometrical
genus 35. Its spectrum consists of the 16 different rationals


${3 \over 7}, {4 \over 7}, {5 \over 7}, {6 \over 7}, {1 \over 1},
{8 \over 7}, {9 \over 7}, {10 \over 7}, {11 \over 7}, {12 \over 7},
{13 \over 7}, {2 \over 1}, {15 \over 7}, {16 \over 7}, {17 \over 7},
{18 \over 7}$

appearing with multiplicities

1,3,6,10,15,21,25,27,27,25,21,15,10,6,3,1.

The singularities of type $\tilde{E}_6$ form a
$\mu$-constant one parameter family given by
$x^3+y^3+z^3+\lambda xyz=0,\quad \lambda^3\neq-27$.
Therefore they have all the same spectrum, which we compute
for 
$x^3+y^3+z^3$.

poly g=x^3+y^3+z^3;
list s2=spectrumnd(g);
s2;
==> [1]:
==>    8
==> [2]:
==>    1
==> [3]:
==>    4
==> [4]:
==>    1,4,5,2
==> [5]:
==>    1,3,3,1
==> [6]:
==>    1,3,3,1
Evaluating semicontinuity is very easy:
semicont(s1,s2);
==> 18

This tells us that there are at most 18 singularities of type
$\tilde{E}_6$ on a septic in $P^3$. But $x^7+y^7+z^7$
is semiquasihomogeneous (sqh), so we can also apply the stronger
form of semicontinuity:

semicontsqh(s1,s2);
==> 17

So in fact a septic has at most 17 triple points of type
$\tilde{E}_6$.

Note that spectrumnd(f) works only if 
$f$
 has nondegenerate
principal part. In fact spectrumnd will detect a degenerate
principal part in many cases and print out an error message.
However if it is known in advance that 
$f$
 has nondegenerate
principal part, then the spectrum may be computed much faster
using spectrumnd(f,1).

<font size="-1">
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; User manual for <a href="http://www.singular.uni-kl.de/"><i>Singular</i></a> version 2-0-4, October 2002,
generated by <a href="http://www.gnu.org/software/texinfo/"><i>texi2html</i></a>.
</font>

</body>
</html>
