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Cohomology Classes

Constructors

General constructors

cohomology_class — Method
cohomology_class(v::NormalToricVarietyType, p::MPolyQuoRingElem)

Construct the toric cohomology class on the toric variety v corresponding to the polynomial p. Note that p must reside in the cohomology ring of v.

Examples

julia> P2 = projective_space(NormalToricVariety, 2)
Normal toric variety

julia> c = cohomology_class(P2, gens(cohomology_ring(P2))[1])
Cohomology class on a normal toric variety given by x1
source
cohomology_class — Method
cohomology_class(d::ToricDivisor)

Construct the toric cohomology class corresponding to the toric divisor d.

Examples

julia> P2 = projective_space(NormalToricVariety, 2)
Normal toric variety

julia> d = toric_divisor(P2, [1, 2, 3])
Torus-invariant, non-prime divisor on a normal toric variety

julia> cohomology_class(d)
Cohomology class on a normal toric variety given by 6*x3
source
cohomology_class — Method
cohomology_class(c::ToricDivisorClass)

Construct the toric cohomology class corresponding to the toric divisor class c.

Examples

julia> P2 = projective_space(NormalToricVariety, 2)
Normal toric variety

julia> tdc = toric_divisor_class(P2, [2])
Divisor class on a normal toric variety

julia> cohomology_class(tdc)
Cohomology class on a normal toric variety given by 2*x3
source
cohomology_class — Method
cohomology_class(l::ToricLineBundle)

Construct the toric cohomology class corresponding to the toric line bundle l.

Examples

julia> P2 = projective_space(NormalToricVariety, 2)
Normal toric variety

julia> l = toric_line_bundle(P2, [2])
Toric line bundle on a normal toric variety

julia> polynomial(cohomology_class(l))
2*x3
source

Addition, subtraction and scalar multiplication

Cohomology classes can be added and subtracted via the usual + and - operators. Moreover, multiplication by scalars from the left is supported for scalars which are integers or of type ZZRingElem or QQFieldElem.

Wedge product

The wedge product of cohomology classes is implemented via *, using internally the multiplication of the corresponding polynomial (equivalence classes) in the Cox ring.

A cohomology class can be wedged n-times with itself via ^n, where n can be an integer or of type ZZRingElem.

Properties

One can check if a cohomology class is trivial via is_trivial.

Equality of cohomology classes can be tested via ==.

Attributes

toric_variety — Method
toric_variety(c::CohomologyClass)

Return the normal toric variety of the cohomology class c.

Examples

julia> dP2 = del_pezzo_surface(NormalToricVariety, 2)
Normal toric variety

julia> d = toric_divisor(dP2, [1, 2, 3, 4, 5])
Torus-invariant, non-prime divisor on a normal toric variety

julia> cc = cohomology_class(d)
Cohomology class on a normal toric variety given by 6*x3 + e1 + 7*e2

julia> toric_variety(cc)
Normal, simplicial, complete toric variety
source
coefficients — Method
coefficients(c::CohomologyClass)

Return the coefficients of the cohomology class c.

Examples

julia> dP2 = del_pezzo_surface(NormalToricVariety, 2)
Normal toric variety

julia> d = toric_divisor(dP2, [1, 2, 3, 4, 5])
Torus-invariant, non-prime divisor on a normal toric variety

julia> cc = cohomology_class(d)
Cohomology class on a normal toric variety given by 6*x3 + e1 + 7*e2

julia> coefficients(cc)
3-element Vector{QQFieldElem}:
 6
 1
 7
source
exponents — Method
exponents(c::CohomologyClass)

Return the exponents of the cohomology class c.

Examples

julia> dP2 = del_pezzo_surface(NormalToricVariety, 2)
Normal toric variety

julia> d = toric_divisor(dP2, [1, 2, 3, 4, 5])
Torus-invariant, non-prime divisor on a normal toric variety

julia> cc = cohomology_class(d)
Cohomology class on a normal toric variety given by 6*x3 + e1 + 7*e2

julia> exponents(cc)
[0   0   1   0   0]
[0   0   0   1   0]
[0   0   0   0   1]
source
polynomial — Method
polynomial(c::CohomologyClass)

Return the polynomial in the cohomology ring of the normal toric variety toric_variety(c) which corresponds to c.

Examples

julia> dP2 = del_pezzo_surface(NormalToricVariety, 2)
Normal toric variety

julia> d = toric_divisor(dP2, [1, 2, 3, 4, 5])
Torus-invariant, non-prime divisor on a normal toric variety

julia> cc = cohomology_class(d)
Cohomology class on a normal toric variety given by 6*x3 + e1 + 7*e2

julia> polynomial(cc)
6*x3 + e1 + 7*e2
source
polynomial — Method
polynomial(c::CohomologyClass, ring::MPolyQuoRing)

Return the polynomial in ring corresponding to the cohomology class c.

Examples

julia> dP2 = del_pezzo_surface(NormalToricVariety, 2)
Normal toric variety

julia> d = toric_divisor(dP2, [1, 2, 3, 4, 5])
Torus-invariant, non-prime divisor on a normal toric variety

julia> cc = cohomology_class(d)
Cohomology class on a normal toric variety given by 6*x3 + e1 + 7*e2

julia> R, _ = polynomial_ring(QQ, 5)
(Multivariate polynomial ring in 5 variables over QQ, QQMPolyRingElem[x1, x2, x3, x4, x5])

julia> (x1, x2, x3, x4, x5) = gens(R)
5-element Vector{QQMPolyRingElem}:
 x1
 x2
 x3
 x4
 x5

julia> sr_and_linear_relation_ideal = ideal([x1*x3, x1*x5, x2*x4, x2*x5, x3*x4, x1 + x2 - x5, x2 + x3 - x4 - x5])
Ideal generated by
  x1*x3
  x1*x5
  x2*x4
  x2*x5
  x3*x4
  x1 + x2 - x5
  x2 + x3 - x4 - x5

julia> R_quo = quo(R, sr_and_linear_relation_ideal)[1]
Quotient
  of multivariate polynomial ring in 5 variables x1, x2, x3, x4, x5
    over rational field
  by ideal (x1*x3, x1*x5, x2*x4, x2*x5, x3*x4, x1 + x2 - x5, x2 + x3 - x4 - x5)

julia> polynomial(R_quo, cc)
6*x3 + x4 + 7*x5
source

Methods

integrate — Method
integrate(c::CohomologyClass; check::Bool = true)

Integrate the cohomolgy class c over the normal toric variety toric_variety(c).

The theory underlying this method requires that the toric variety in question is simplicial and complete. The check of completeness may take a long time to complete. If desired, this can be switched off by setting the optional argument check to the value false.

Examples

julia> dP3 = del_pezzo_surface(NormalToricVariety, 3)
Normal toric variety

julia> (x1, x2, x3, e1, e2, e3) = gens(cohomology_ring(dP3))
6-element Vector{MPolyQuoRingElem{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}}:
 x1
 x2
 x3
 e1
 e2
 e3

julia> c = cohomology_class(dP3, e3*e3 + e3)
Cohomology class on a normal toric variety given by e3^2 + e3

julia> integrate(c)
-1

julia> F3 = hirzebruch_surface(NormalToricVariety, 3)
Normal toric variety

julia> (x1, x2, x3, x4) = gens(cohomology_ring(F3))
4-element Vector{MPolyQuoRingElem{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}}:
 t1
 x1
 t2
 x2

julia> c = cohomology_class(F3, x1*x2 + x3*x4)
Cohomology class on a normal toric variety given by 2//3*x2^2

julia> integrate(c)
2

The following example constructs the Fano variety 2-36 (cf. https://www.fanography.info/2-36) and verifies that the triple self-intersection number of its anticanonical bundle is 62.

Examples

julia> e1 = [1,0,0];

julia> e2 = [0,1,0];

julia> e3 = [0,0,1];

julia> m = 2;

julia> ray_generators = [e1, -e1, e2, e3, - e2 - e3 - m * e1];

julia> max_cones = incidence_matrix([[1,3,4], [1,3,5], [1,4,5], [2,3,4], [2,3,5], [2,4,5]]);

julia> X = normal_toric_variety(max_cones, ray_generators; non_redundant = true)
Normal toric variety

julia> cox_ring(X)
Multivariate polynomial ring in 5 variables over QQ graded by
  x1 -> [1 0]
  x2 -> [1 2]
  x3 -> [0 -1]
  x4 -> [0 -1]
  x5 -> [0 -1]

julia> cohomology_ring(X)
Quotient
  of multivariate polynomial ring in 5 variables over QQ graded by
    x1 -> [1]
    x2 -> [1]
    x3 -> [1]
    x4 -> [1]
    x5 -> [1]
  by ideal (x1 - x2 - 2*x5, x3 - x5, x4 - x5, x1*x2, x3*x4*x5)

julia> integrate(cohomology_class(anticanonical_divisor(X))^3)
62

julia> integrate(cohomology_class(anticanonical_divisor_class(X))^3)
62
source

Special attributes of toric varieties

cohomology_ring — Method
cohomology_ring(v::NormalToricVarietyType; check::Bool = true)

Return the cohomology ring of the simplicial and complete toric variety v.

Examples

julia> p2 = projective_space(NormalToricVariety, 2);

julia> ngens(cohomology_ring(p2))
3
source
volume_form — Method
volume_form(v::NormalToricVariety)

Construct the volume form of the normal toric toric variety v.

Examples

julia> polynomial(volume_form(projective_space(NormalToricVariety, 2)))
x3^2

julia> polynomial(volume_form(del_pezzo_surface(NormalToricVariety, 3)))
-e3^2

julia> polynomial(volume_form(hirzebruch_surface(NormalToricVariety, 5)))
1//5*x2^2
source
intersection_form — Method
intersection_form(v::NormalToricVariety)

Computes the intersection numbers among the cohomology classes associated to the torusinvariant prime divisors of the normal toric toric variety v.

Examples

julia> F3 = hirzebruch_surface(NormalToricVariety, 3)
Normal toric variety

julia> length(intersection_form(F3))
10
source
chern_class — Method
chern_class(v::NormalToricVariety, k::Int; check::Bool = true)

Computes the k-th Chern class of the tangent bundle of a normal toric variety that is both smooth and complete. Since these checks can be computationally very demanding, we provide an optional argument check. Once set to false, this method skips those tests.

The implemented algorithm uses proposition 13.1.2 in [CLS11].

Examples

julia> F3 = hirzebruch_surface(NormalToricVariety, 3)
Normal toric variety

julia> chern_class(F3, 0)
Cohomology class on a normal toric variety given by 1

julia> chern_class(F3, 1, check = false)
Cohomology class on a normal toric variety given by t1 + x1 + t2 + x2

julia> integrate(chern_class(F3, 2), check = false)
4
source
chern_classes — Method
chern_classes(v::NormalToricVariety; check::Bool = true)

Computes all Chern classes of the tangent bundle of a normal toric variety, which is smooth and complete. Since those checks can be computationally very demanding, the optional argument check can be set to false to skip those tests.

Examples

julia> F3 = hirzebruch_surface(NormalToricVariety, 3)
Normal toric variety

julia> cs = chern_classes(F3)
3-element Vector{CohomologyClass}:
 Cohomology class on a normal toric variety given by 1
 Cohomology class on a normal toric variety given by t1 + x1 + t2 + x2
 Cohomology class on a normal toric variety given by 4//3*x2^2

julia> integrate(cs[3])
4
source
basis_of_h4 — Method
basis_of_h4(v::NormalToricVariety; check::Bool = true)

This method computes a monomial basis of the cohomology class $H^4(X, \mathbb{Q})$ for a toric variety $X$. The algorithm employs Theorem 12.4.1 in [CLS11], i.e. truncates the cohomology ring to degree $2$. By virtue of this theorem, this approach is supported only for toric varieties that are both complete and simplicial. Since it can be computationally very demanding to verify completeness, the optional argument check can be set to false to skip the tests.

Examples

julia> Y1 = hirzebruch_surface(NormalToricVariety, 2)
Normal toric variety

julia> Y2 = hirzebruch_surface(NormalToricVariety, 2)
Normal toric variety

julia> Y = Y1 * Y2
Normal toric variety

julia> h4_basis = basis_of_h4(Y)
6-element Vector{CohomologyClass}:
 Cohomology class on a normal toric variety given by yx2^2
 Cohomology class on a normal toric variety given by xx2*yx2
 Cohomology class on a normal toric variety given by xx2*yt2
 Cohomology class on a normal toric variety given by xx2^2
 Cohomology class on a normal toric variety given by xt2*yx2
 Cohomology class on a normal toric variety given by xt2*yt2

julia> betti_number(Y, 4) == length(h4_basis)
true
source