Given a toric divisor D, or its degree d, its cohomology vector is the list $\{h^0(X, OO_X(d)), h^1(X, OO_X(d)), \ldots, h^{dim X}(X, OO_X(d)) \}$.
Here is an example.
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
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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
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i7 : cohomCalg(D, Silent => true)
o7 = {0, 0, 0, 194, 0}
o7 : List
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We compare this to the results from NormalToricVarieties.
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
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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}}}}
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The object cohomCalg is a method function with options.