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C.5 Gauss-Manin connection
Let f:(Cn+1,0) → (C,0) be a
complex isolated hypersurface singularity given by a polynomial with algebraic coefficients which we also
denote by f. Let O = C[x0,…,xn](x0,…,xn) be the local ring at the origin and Jf the Jacobian ideal of
f.
A Milnor representative of f defines a differentiable fibre bundle over the punctured disc with fibres of
homotopy type of μ n-spheres. The n-th cohomology bundle is a flat vector bundle of dimension n and carries a
natural flat connection with covariant derivative ∂t. The 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 moderate growth at 0 form a regular D = C{t}[∂t]-module G, the 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 Brieskorn lattice
H“ := ΩCn+1,0n+1∕df ∧ dΩCn+1,0n−1 in G.
The D-module structure defines the V-filtration V on G by V α := ∑
β≥αC{t}ker(t∂t −β)n+1. The Brieskorn
lattice defines the Hodge filtration F on G by Fk = ∂tkH“ which comes from the mixed Hodge structure on
the Milnor fibre. Note that F−1 = H‘.
The induced V-filtration on the Brieskorn lattice determines the singularity spectrum Sp by
Sp(α) := dimCGrV αGr0FG. The spectrum consists of μ rational numbers α1,…,αμ such that e2πiα1,…,e2πiαμ are
the eigenvalues of the monodromy. These spectral numbers lie in the open interval (−1,n), symmetric about the
midpoint (n − 1)∕2.
The spectrum is constant under μ-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 ⊂ Pn 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 μ-constant deformation of x0d + … + xnd = 0, the singularities are bounded by the spectrum of
x0d + … + xnd.
Using the library gaussman.lib one can compute the monodromy, the V-filtration on H“∕H‘, and the
spectrum.
Let us consider as an example
f = x5 + x2y2 + y5
.
First, we compute a matrix
M
such that
exp(2πiM)
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∕Jf H“∕H‘
in l[4] .
If the principal
part of f is C-nondegenerate, one can compute the spectrum using the library 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
Ẽ6 on a surface X ⊂ P3
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
μ(f), the geometric genus pg(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
x7 + y7 + z7 = 0
has Milnor number 216 and geometrical
genus 35. Its spectrum consists of the 16 different rationals
3_
7, 4
7, 5
7, 6
7, 1
1, 8
7, 9
7, 10
7 , 11
7 , 12
7 , 13
7 , 2
1, 15
7 , 16
7 , 17
7 , 18
7
appearing with multiplicities
1,3,6,10,15,21,25,27,27,25,21,15,10,6,3,1.
The singularities of type Ẽ6 form a μ-constant one parameter family
given by x3 + y3 + z3 + λxyz = 0, λ3≠− 27.
Therefore they have all the same spectrum, which we compute
for
x3 + y3 + z3.
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:
This tells us that there are at most 18 singularities of type
Ẽ6 on a septic in P3. But x7 + y7 + z7
is semiquasihomogeneous (sqh), so we can also apply the stronger
form of semicontinuity:
So in fact a septic has at most 17 triple points of type
Ẽ6.
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) .
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