cohomCalg -- compute cohomology vectors using the CohomCalg software

Synopsis

• Usage:

  cvecs = cohomCalg(X, L)

  cvecs = cohomCalg D

• Inputs:
  • X, a normal toric variety,
  • L, a list, of toric divisors, or degrees, or a single degree, or a toric divisor
  • D, a toric divisor, a single toric divisor
• Optional inputs:
  • Silent => a Boolean value, default value null, but if null, then the default value from the configuration file is overridden by this value. If true, then all output from CohomCalg is suppressed.
• Outputs:
  • cvecs, a list, The list of cohomology vectors of the toric divisors in L (or a single list, if the input was a single degree or toric divisor)
• Consequences:

  • The answers are stashed into the mutable hashtable X.cache#CohomCalg, and are not recomputed if possible.

Description

i1 : needsPackage "ReflexivePolytopesDB"

o1 = ReflexivePolytopesDB

o1 : Package

i2 : topes = kreuzerSkarke(5, Limit => 20);
using offline data file: ks5-n50.txt

i3 : A = matrix topes_15

o3 = | 1 1 0 1 -1 -2 1  |
     | 0 2 0 0 -4 0  6  |
     | 0 0 1 0 2  -1 -4 |
     | 0 0 0 2 -2 0  0  |

              4       7
o3 : Matrix ZZ  <-- ZZ

i4 : P = convexHull A

o4 = P

o4 : Polyhedron

i5 : X = normalToricVariety P

o5 = X

o5 : NormalToricVariety

i6 : D = 3 * X_0 - 5 * X_4

o6 = 3*X  - 5*X
        0      4

o6 : ToricDivisor on X

i7 : cohomCalg(D, Silent => true)

o7 = {0, 0, 0, 194, 0}

o7 : List

i8 : cohomCalg(X, D)

o8 = {0, 0, 0, 194, 0}

o8 : List

i9 : for i from 0 to dim X list rank HH^i(X, OO D)

o9 = {0, 0, 0, 194, 0}

o9 : List

i10 : peek cohomCalg X

o10 = MutableHashTable{{-22, 3, 17} => {{0, 0, 0, 194, 0}, {{3,
      -----------------------------------------------------------------------
      1x1*x2*x4*x5}, {3, 1x1*x2*x3*x4*x5}, {3, 1x1*x2*x4*x5*x6}}}}

See also

• cohomCalg(NormalToricVariety) -- locally stashed cohomology vectors from CohomCalg
• ReflexivePolytopesDB -- simple access to Kreuzer-Skarke database of reflexive polytopes of dimensions 3 and 4