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CohomCalg -- an interface to the CohomCalg software for computing cohomology of torus invariant divisors on a toric variety

Description

CohomCalg is software written by Benjamin Jurke and Thorsten Rahn (in collaboration with Ralph Blumenhagen and Helmut Roschy) for computing the cohomology vectors of torus invariant divisors on a (normal) toric variety (see https://github.com/BenjaminJurke/cohomCalg for more information).

CohomCalg is an efficient and careful implementation. One limitation is that the number of rays in the fan and the number of generators of the Stanley-Reisner ideal of the fan must both be no larger than 64.

Here is a sample usage of this package in Macaulay2. Let's compute the cohomology of some divisors on a smooth Fano toric variety.

i1 : needsPackage "NormalToricVarieties"

o1 = NormalToricVarieties

o1 : Package
 
i2 : X = smoothFanoToricVariety(3,15)

o2 = X

o2 : NormalToricVariety
 
i3 : rays X

o3 = {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}, {0, -1, -1}, {0, -1, 0}, {-1, 0, 0},
     ------------------------------------------------------------------------
     {-1, 1, 0}}

o3 : List
 
i4 : max X

o4 = {{0, 1, 2}, {0, 1, 3}, {0, 2, 4}, {0, 3, 4}, {1, 2, 6}, {1, 3, 6}, {2,
     ------------------------------------------------------------------------
     4, 5}, {2, 5, 6}, {3, 4, 5}, {3, 5, 6}}

o4 : List
 
i5 : S = ring X

o5 = S

o5 : PolynomialRing
 
i6 : SR = dual monomialIdeal X

o6 = monomialIdeal (x x , x x , x x , x x , x x , x x )
                     2 3   1 4   0 5   1 5   0 6   4 6

o6 : MonomialIdeal of S
 
i7 : KX = toricDivisor X

o7 = - X  - X  - X  - X  - X  - X  - X
        0    1    2    3    4    5    6

o7 : ToricDivisor on X
 
i8 : assert isVeryAmple (-KX)
 
i9 : cohoms1 = for i from 0 to 6 list X_i => cohomCalg X_i

o9 = {X  => {2, 0, 0, 0}, X  => {1, 0, 0, 0}, X  => {2, 0, 0, 0}, X  => {2,
       0                   1                   2                   3       
     ------------------------------------------------------------------------
     0, 0, 0}, X  => {1, 0, 0, 0}, X  => {1, 0, 0, 0}, X  => {1, 0, 0, 0}}
                4                   5                   6

o9 : List
 
i10 : cohoms2 = for i from 0 to 6  list X_i => (
          for j from 0 to dim X list rank HH^j(X, OO_X(toSequence degree X_i))
          )

o10 = {X  => {2, 0, 0, 0}, X  => {1, 0, 0, 0}, X  => {2, 0, 0, 0}, X  => {2,
        0                   1                   2                   3       
      -----------------------------------------------------------------------
      0, 0, 0}, X  => {1, 0, 0, 0}, X  => {1, 0, 0, 0}, X  => {1, 0, 0, 0}}
                 4                   5                   6

o10 : List
 
i11 : assert(cohoms1 === cohoms2)

For efficiency reasons, it is better, if this works for your use, to call CohomCalg by batching together several cohomology requests.

i12 : needsPackage "ReflexivePolytopesDB"

o12 = ReflexivePolytopesDB

o12 : Package
 
i13 : topes = kreuzerSkarke(21, Limit => 20);
using offline data file: ks21-n100.txt
 
i14 : A = matrix topes_10

o14 = | 1 0 0 -1 2  0  0 -3 -2 1  |
      | 0 1 0 1  -1 1  0 1  0  -1 |
      | 0 0 1 1  -1 -1 0 4  2  -2 |
      | 0 0 0 0  0  0  1 -1 -1 1  |

               4       10
o14 : Matrix ZZ  <-- ZZ
 
i15 : P = convexHull A

o15 = P

o15 : Polyhedron
 
i16 : X = normalToricVariety P

o16 = X

o16 : NormalToricVariety
 
i17 : SR = dual monomialIdeal X

o17 = monomialIdeal (x x , x x x , x x , x x x , x x x , x x x x , x x x ,
                      1 2   0 1 3   0 4   0 2 6   0 3 6   1 3 5 6   1 3 7 
      -----------------------------------------------------------------------
      x x x , x x x x , x x x , x x x , x x x x , x x x x , x x x , x x x x ,
       1 4 7   0 3 5 7   2 4 8   2 6 8   3 5 6 8   4 5 6 8   4 7 8   2 5 7 8 
      -----------------------------------------------------------------------
      x x x x , x x x x , x x x , x x x , x x , x x x , x x x , x x x ,
       3 5 7 8   3 6 7 8   0 1 9   2 4 9   5 9   0 6 9   2 6 9   1 7 9 
      -----------------------------------------------------------------------
      x x x )
       4 7 9

o17 : MonomialIdeal of QQ[x ..x ]
                           0   9
 
i18 : D2 = subsets(for i from 0 to #rays X - 1 list (-X_i), 2)

o18 = {{- X , - X }, {- X , - X }, {- X , - X }, {- X , - X }, {- X , - X },
           0     1       0     2       1     2       0     3       1     3  
      -----------------------------------------------------------------------
      {- X , - X }, {- X , - X }, {- X , - X }, {- X , - X }, {- X , - X },
          2     3       0     4       1     4       2     4       3     4  
      -----------------------------------------------------------------------
      {- X , - X }, {- X , - X }, {- X , - X }, {- X , - X }, {- X , - X },
          0     5       1     5       2     5       3     5       4     5  
      -----------------------------------------------------------------------
      {- X , - X }, {- X , - X }, {- X , - X }, {- X , - X }, {- X , - X },
          0     6       1     6       2     6       3     6       4     6  
      -----------------------------------------------------------------------
      {- X , - X }, {- X , - X }, {- X , - X }, {- X , - X }, {- X , - X },
          5     6       0     7       1     7       2     7       3     7  
      -----------------------------------------------------------------------
      {- X , - X }, {- X , - X }, {- X , - X }, {- X , - X }, {- X , - X },
          4     7       5     7       6     7       0     8       1     8  
      -----------------------------------------------------------------------
      {- X , - X }, {- X , - X }, {- X , - X }, {- X , - X }, {- X , - X },
          2     8       3     8       4     8       5     8       6     8  
      -----------------------------------------------------------------------
      {- X , - X }, {- X , - X }, {- X , - X }, {- X , - X }, {- X , - X },
          7     8       0     9       1     9       2     9       3     9  
      -----------------------------------------------------------------------
      {- X , - X }, {- X , - X }, {- X , - X }, {- X , - X }, {- X , - X }}
          4     9       5     9       6     9       7     9       8     9

o18 : List
 
i19 : D2 = D2/sum/degree

o19 = {{0, 1, -2, -2, 4, 0}, {0, 1, -1, 1, 0, -2}, {2, 2, 3, 1, -4, -6}, {-1,
      -----------------------------------------------------------------------
      0, -4, -2, 5, 3}, {1, 1, 0, -2, 1, -1}, {1, 1, 1, 1, -3, -3}, {-2, 0,
      -----------------------------------------------------------------------
      -3, -1, 4, 2}, {0, 1, 1, -1, 0, -2}, {0, 1, 2, 2, -4, -4}, {-1, 0, -1,
      -----------------------------------------------------------------------
      -1, 1, 1}, {-1, -1, -3, -1, 4, 2}, {1, 0, 1, -1, 0, -2}, {1, 0, 2, 2,
      -----------------------------------------------------------------------
      -4, -4}, {0, -1, -1, -1, 1, 1}, {-1, -1, 0, 0, 0, 0}, {-1, 0, -4, -1,
      -----------------------------------------------------------------------
      4, 2}, {1, 1, 0, -1, 0, -2}, {1, 1, 1, 2, -4, -4}, {0, 0, -2, -1, 1,
      -----------------------------------------------------------------------
      1}, {-1, 0, -1, 0, 0, 0}, {0, -1, -1, 0, 0, 0}, {-1, 0, -3, -2, 4, 2},
      -----------------------------------------------------------------------
      {1, 1, 1, -2, 0, -2}, {1, 1, 2, 1, -4, -4}, {0, 0, -1, -2, 1, 1}, {-1,
      -----------------------------------------------------------------------
      0, 0, -1, 0, 0}, {0, -1, 0, -1, 0, 0}, {0, 0, -1, -1, 0, 0}, {-1, 0,
      -----------------------------------------------------------------------
      -3, -1, 3, 2}, {1, 1, 1, -1, -1, -2}, {1, 1, 2, 2, -5, -4}, {0, 0, -1,
      -----------------------------------------------------------------------
      -1, 0, 1}, {-1, 0, 0, 0, -1, 0}, {0, -1, 0, 0, -1, 0}, {0, 0, -1, 0,
      -----------------------------------------------------------------------
      -1, 0}, {0, 0, 0, -1, -1, 0}, {-1, 0, -3, -1, 4, 1}, {1, 1, 1, -1, 0,
      -----------------------------------------------------------------------
      -3}, {1, 1, 2, 2, -4, -5}, {0, 0, -1, -1, 1, 0}, {-1, 0, 0, 0, 0, -1},
      -----------------------------------------------------------------------
      {0, -1, 0, 0, 0, -1}, {0, 0, -1, 0, 0, -1}, {0, 0, 0, -1, 0, -1}, {0,
      -----------------------------------------------------------------------
      0, 0, 0, -1, -1}}

o19 : List
 
i20 : elapsedTime hvecs = cohomCalg(X, D2)
 -- 3.17828 seconds elapsed

o20 = {{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}, {0, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 0, 0, 0}, {0,
      -----------------------------------------------------------------------
      0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0,
      -----------------------------------------------------------------------
      0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0,
      -----------------------------------------------------------------------
      0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0,
      -----------------------------------------------------------------------
      0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0},
      -----------------------------------------------------------------------
      {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0,
      -----------------------------------------------------------------------
      0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0,
      -----------------------------------------------------------------------
      0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0,
      -----------------------------------------------------------------------
      0, 0}, {0, 0, 0, 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}, {0, 0, 0, 0, 0}}

o20 : List
 
i21 : peek cohomCalg X

o21 = MutableHashTable{{-1, -1, -3, -1, 4, 2} => {{0, 0, 0, 0, 0}, {}}          }
                       {-1, -1, 0, 0, 0, 0} => {{0, 0, 0, 0, 0}, {}}
                       {-1, 0, -1, -1, 1, 1} => {{0, 0, 0, 0, 0}, {}}
                       {-1, 0, -1, 0, 0, 0} => {{0, 0, 0, 0, 0}, {}}
                       {-1, 0, -3, -1, 3, 2} => {{0, 0, 0, 0, 0}, {}}
                       {-1, 0, -3, -1, 4, 1} => {{0, 0, 0, 0, 0}, {}}
                       {-1, 0, -3, -2, 4, 2} => {{0, 0, 0, 0, 0}, {}}
                       {-1, 0, -4, -1, 4, 2} => {{0, 0, 0, 0, 0}, {}}
                       {-1, 0, -4, -2, 5, 3} => {{0, 0, 0, 0, 0}, {}}
                       {-1, 0, 0, -1, 0, 0} => {{0, 0, 0, 0, 0}, {}}
                       {-1, 0, 0, 0, -1, 0} => {{0, 0, 0, 0, 0}, {}}
                       {-1, 0, 0, 0, 0, -1} => {{0, 0, 0, 0, 0}, {}}
                       {-2, 0, -3, -1, 4, 2} => {{0, 1, 0, 0, 0}, {{1, 1x0*x4}}}
                       {0, -1, -1, -1, 1, 1} => {{0, 0, 0, 0, 0}, {}}
                       {0, -1, -1, 0, 0, 0} => {{0, 0, 0, 0, 0}, {}}
                       {0, -1, 0, -1, 0, 0} => {{0, 0, 0, 0, 0}, {}}
                       {0, -1, 0, 0, -1, 0} => {{0, 0, 0, 0, 0}, {}}
                       {0, -1, 0, 0, 0, -1} => {{0, 1, 0, 0, 0}, {{1, 1x5*x9}}}
                       {0, 0, -1, -1, 0, 0} => {{0, 0, 0, 0, 0}, {}}
                       {0, 0, -1, -1, 0, 1} => {{0, 0, 0, 0, 0}, {}}
                       {0, 0, -1, -1, 1, 0} => {{0, 0, 0, 0, 0}, {}}
                       {0, 0, -1, -2, 1, 1} => {{0, 0, 0, 0, 0}, {}}
                       {0, 0, -1, 0, -1, 0} => {{0, 0, 0, 0, 0}, {}}
                       {0, 0, -1, 0, 0, -1} => {{0, 0, 0, 0, 0}, {}}
                       {0, 0, -2, -1, 1, 1} => {{0, 0, 0, 0, 0}, {}}
                       {0, 0, 0, -1, -1, 0} => {{0, 0, 0, 0, 0}, {}}
                       {0, 0, 0, -1, 0, -1} => {{0, 0, 0, 0, 0}, {}}
                       {0, 0, 0, 0, -1, -1} => {{0, 0, 0, 0, 0}, {}}
                       {0, 1, -1, 1, 0, -2} => {{0, 0, 0, 0, 0}, {}}
                       {0, 1, -2, -2, 4, 0} => {{0, 0, 0, 0, 0}, {}}
                       {0, 1, 1, -1, 0, -2} => {{0, 0, 0, 0, 0}, {}}
                       {0, 1, 2, 2, -4, -4} => {{0, 0, 0, 0, 0}, {}}
                       {1, 0, 1, -1, 0, -2} => {{0, 0, 0, 0, 0}, {}}
                       {1, 0, 2, 2, -4, -4} => {{0, 0, 0, 0, 0}, {}}
                       {1, 1, 0, -1, 0, -2} => {{0, 0, 0, 0, 0}, {}}
                       {1, 1, 0, -2, 1, -1} => {{0, 0, 0, 0, 0}, {}}
                       {1, 1, 1, -1, -1, -2} => {{0, 0, 0, 0, 0}, {}}
                       {1, 1, 1, -1, 0, -3} => {{0, 0, 0, 0, 0}, {}}
                       {1, 1, 1, -2, 0, -2} => {{0, 0, 0, 0, 0}, {}}
                       {1, 1, 1, 1, -3, -3} => {{0, 0, 0, 0, 0}, {}}
                       {1, 1, 1, 2, -4, -4} => {{0, 0, 0, 0, 0}, {}}
                       {1, 1, 2, 1, -4, -4} => {{0, 0, 0, 0, 0}, {}}
                       {1, 1, 2, 2, -4, -5} => {{0, 0, 0, 0, 0}, {}}
                       {1, 1, 2, 2, -5, -4} => {{0, 0, 0, 0, 0}, {}}
                       {2, 2, 3, 1, -4, -6} => {{0, 1, 0, 0, 0}, {{1, 1x1*x2}}}
 
i22 : degree(X_3 + X_7 + X_8)

o22 = {0, 0, 1, 2, 0, -1}

o22 : List
 
i23 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 + X_8)
 -- 0.402505 seconds elapsed

o23 = {1, 0, 0, 0, 0}

o23 : List
 
i24 : elapsedTime cohomvec2 = for j from 0 to dim X list rank HH^j(X, OO_X(0,0,1,2,0,-1))
 -- 8.74977 seconds elapsed

o24 = {1, 0, 0, 0, 0}

o24 : List
 
i25 : assert(cohomvec1 == cohomvec2)
 
i26 : degree(X_3 + X_7 - X_8)

o26 = {0, 0, 1, 2, -2, -1}

o26 : List
 
i27 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 - X_8)
 -- 0.408693 seconds elapsed

o27 = {0, 0, 0, 0, 0}

o27 : List
 
i28 : elapsedTime cohomvec2 = elapsedTime for j from 0 to dim X list rank HH^j(X, OO_X(0,0,1,2,-2,-1))
 -- 0.40388 seconds elapsed
 -- 0.403903 seconds elapsed

o28 = {0, 0, 0, 0, 0}

o28 : List
 
i29 : assert(cohomvec1 == cohomvec2)

cohomCalg computes cohomology vectors by calling CohomCalg. It also stashes it's results in the toric variety's cache table, so computations need not be performed twice.

See also

Author

Version

This documentation describes version 0.8 of CohomCalg.

Source code

The source code from which this documentation is derived is in the file CohomCalg.m2. The auxiliary files accompanying it are in the directory CohomCalg/.

Exports

  • Functions and commands
    • cohomCalg -- compute cohomology vectors using the CohomCalg software
  • Methods
    • cohomCalg(NormalToricVariety,List) -- see cohomCalg -- compute cohomology vectors using the CohomCalg software
    • cohomCalg(NormalToricVariety,ToricDivisor) -- see cohomCalg -- compute cohomology vectors using the CohomCalg software
    • cohomCalg(ToricDivisor) -- see cohomCalg -- compute cohomology vectors using the CohomCalg software
    • cohomCalg(NormalToricVariety) -- locally stashed cohomology vectors from CohomCalg
  • Symbols
    • Silent (missing documentation)

For the programmer

The object CohomCalg is a package.