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)
 -- 2.52687s 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)
 -- .331754s 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))
 -- 7.11834s 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)
 -- .347141s 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))
 -- .26071s elapsed
 -- .260738s 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.

Author

Michael E. Stillman <[email protected]>

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.