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C.5 Gauss-Manin connection
Let
702#702 be a complex isolated hypersurface singularity given by a polynomial with algebraic coefficients which we also denote by 267#267.
Let
703#703 be the local ring at the origin and 704#704 the Jacobian ideal of 267#267.
A Milnor representative of 267#267 defines a differentiable fibre bundle over the punctured disc with fibres of homotopy type of 705#705 17#17-spheres.
The 17#17-th cohomology bundle is a flat vector bundle of dimension 17#17 and carries a natural flat connection with covariant derivative 706#706.
The monodromy operator is the action of a positively oriented generator of the fundamental group of the punctured disc on the Milnor fibre.
Sections in the cohomology bundle of moderate growth at 2#2 form a regular
707#707-module 190#190, the Gauss-Manin connection.
By integrating along flat multivalued families of cycles, one can consider fibrewise global holomorphic differential forms as elements of 190#190.
This factors through an inclusion of the Brieskorn lattice
708#708 in 190#190.
The 240#240-module structure defines the V-filtration 324#324 on 190#190 by
709#709.
The Brieskorn lattice defines the Hodge filtration 710#710 on 190#190 by
711#711 which comes from the mixed Hodge structure on the Milnor fibre.
Note that 712#712.
The induced V-filtration on the Brieskorn lattice determines the singularity spectrum 713#713 by
714#714.
The spectrum consists of 705#705 rational numbers
715#715 such that
716#716 are the eigenvalues of the monodromy.
These spectral numbers lie in the open interval 717#717, symmetric about the midpoint 718#718.
The spectrum is constant under 705#705-constant deformations and has the following semicontinuity property:
The number of spectral numbers in an interval 719#719 of all singularities of a small deformation of 267#267 is greater than or equal to that of f in this interval.
For semiquasihomogeneous singularities, this also holds for intervals of the form 720#720.
Two given isolated singularities 267#267 and 149#149 determine two spectra and from these spectra we get an integer.
This integer is the maximal positive integer 282#282 such that the semicontinuity holds for the spectrum of 267#267 and 282#282 times the spectrum of 149#149.
These numbers give bounds for the maximal number of isolated singularities of a specific type on a hypersurface 721#721 of degree 171#171:
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 705#705-constant deformation of
722#722, the singularities are bounded by the spectrum of
723#723.
Using the library gmssing.lib one can compute the monodromy, the V-fitration on 724#724, and the spectrum.
Let us consider as an example
725#725.
First, we compute a matrix
13#13 such that
726#726is a monodromy matrix of
267#267 and the Jordan normal form of
13#13:
| LIB "mondromy.lib";
ring R=0,(x,y),ds;
poly f=x5+x2y2+y5;
matrix M=monodromyB(f);
print(M);
==> 11/10,0, 0, 0, 0, 0,-1/4,0, 0, 0, 0,
==> 0, 13/10,0, 0, 0, 0,0, 15/8,0, 0, 0,
==> 0, 0, 13/10,0, 0, 0,0, 0, 15/8,0, 0,
==> 0, 0, 0, 11/10,-1/4,0,0, 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, 3/5,
==> 0, 0, 0, 0, 0, 0,9/10,0, 0, 0, 0,
==> 0, 0, 0, 0, 0, 0,0, 7/10,0, 0, 0,
==> 0, 0, 0, 0, 0, 0,0, 0, 7/10,0, 0,
==> 0, 0, 0, 0, 0, 0,0, 0, 0, 1, -2/5,
==> 0, 0, 0, 0, 0, 0,0, 0, 0, 5/8,0
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Now, we compute the V-fitration on
724#724 and the spectrum:
| LIB "gmssing.lib";
ring R=0,(x,y),ds;
poly f=x5+x2y2+y5;
list l=vfilt(f);
print(l[1]); // spectral numbers
==> -1/2,
==> -3/10,
==> -1/10,
==> 0,
==> 1/10,
==> 3/10,
==> 1/2
print(l[2]); // corresponding multiplicities
==> 1,
==> 2,
==> 2,
==> 1,
==> 2,
==> 2,
==> 1
print(l[3]); // vector space of i-th graded part
==> [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]); // monomial vector space basis of H''/s*H''
==> y5,
==> y4,
==> y3,
==> y2,
==> xy,
==> y,
==> x4,
==> x3,
==> x2,
==> x,
==> 1
print(l[5]); // standard basis of Jacobian ideal
==> 2x2y+5y4,
==> 5x5-5y5,
==> 2xy2+5x4,
==> 10y6+25x3y4
| Here l[1] contains the spectral numbers, l[2] the corresponding multiplicities, l[3] a
78#78-basis of the V-filtration on
724#724 in terms of the monomial basis of
727#727in l[4] (separated by degree).
If the principal part of 267#267 is 78#78-nondegenerate, one can compute the spectrum using the library spectrum.lib.
In this case, the V-filtration on 728#728 coincides with the Newton-filtration on 728#728 which allows to compute the spectrum more efficiently.
Let us calculate one specific example, the maximal number
of triple points of type
729#729 on a surface 730#730of degree seven.
This calculation can be done over the rationals.
We choose a local ordering on
39#39. Here we take the
negative degree lexicographical ordering, in SINGULAR denoted by ds :
| 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
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The command spectrumnd(f) computes the spectrum of
267#267 and
returns a list with six entries:
The Milnor number
731#731, the geometric genus 732#732and 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
733#733has Milnor number 216 and geometrical
genus 35. Its spectrum consists of the 16 different rationals
734#734 appearing with multiplicities
1,3,6,10,15,21,25,27,27,25,21,15,10,6,3,1.
The singularities of type 729#729 form a
705#705-constant one parameter family given by
735#735.Therefore they have all the same spectrum, which we compute
for
736#736.
| 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
729#729 on a septic in 737#737. But 738#738is semiquasihomogeneous (sqh), so we can also apply the stronger
form of semicontinuity:
| semicontsqh(s1,s2);
==> 17
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So in fact a septic has at most 17 triple points of type
729#729.
Note that spectrumnd(f) works only if
267#267 has a 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
267#267 has nondegenerate
principal part, then the spectrum may be computed much faster
using spectrumnd(f,1) .
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