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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    

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

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:
 
semicont(s1,s2);
==> 18

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

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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