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CohomCalg :: CohomCalg

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.

needsPackage "NormalToricVarieties"
X = smoothFanoToricVariety(3,15)
rays X
max X
S = ring X
SR = dual monomialIdeal X
KX = toricDivisor X
assert isVeryAmple (-KX)
cohoms1 = for i from 0 to 6 list X_i => cohomCalg X_i
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)) )
assert(cohoms1 === cohoms2)

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

needsPackage "ReflexivePolytopesDB"
topes = kreuzerSkarke(21, Limit => 20);
A = matrix topes_10
P = convexHull A
X = normalToricVariety P
SR = dual monomialIdeal X
D2 = subsets(for i from 0 to #rays X - 1 list (-X_i), 2)
D2 = D2/sum/degree
elapsedTime hvecs = cohomCalg(X, D2)
peek cohomCalg X
degree(X_3 + X_7 + X_8)
elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 + X_8)
elapsedTime cohomvec2 = for j from 0 to dim X list rank HH^j(X, OO_X(0,0,1,2,0,-1))
assert(cohomvec1 == cohomvec2)
degree(X_3 + X_7 - X_8)
elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 - X_8)
elapsedTime cohomvec2 = elapsedTime for j from 0 to dim X list rank HH^j(X, OO_X(0,0,1,2,-2,-1))
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.

Exports

  • Functions and commands
    • cohomCalg -- compute cohomology vectors using the CohomCalg software
  • Symbols
    • Silent, see cohomCalg -- compute cohomology vectors using the CohomCalg software
  • Other things