Abstract Variety Maps

map(X::AbstractVariety, Y::AbstractVariety, fˣ::Vector, fₓ = nothing; inclusion::Bool = false, symbol::String = "x")

Return an abstract variety map X $\rightarrow$ Y by specifying the pullbacks of the generators of the Chow ring of Y.

In the case of an inclusion X $\hookrightarrow$ Y where the class of X is not present in the Chow ring of Y, use the argument inclusion = true. Then, a copy of Y will be created, with extra classes added so that one can pushforward all classes on X.

Examples

julia> P2xP2 = abstract_projective_space(2, symbol = "k")*abstract_projective_space(2, symbol = "l")
AbstractVariety of dim 4

julia> k, l = gens(P2xP2)
2-element Vector{MPolyQuoRingElem{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}}:
 k
 l

julia> P8 = abstract_projective_space(8)
AbstractVariety of dim 8

julia> h = gens(P8)[1]
h

julia> Se = map(P2xP2, P8, [k+l]) # Segre embedding
AbstractVarietyMap from AbstractVariety of dim 4 to AbstractVariety of dim 8

julia> pullback(Se, h)
k + l

julia> pushforward(Se, k+l)
6*h^5

identity_map(X::AbstractVariety)

Return the identity map on X.

 domain(f::AbstractVarietyMap)

Return the domain of f.

 codomain(f::AbstractVarietyMap)

Return the codomain of f.

dim(f::AbstractVarietyMap)

Return the relative dimension of f, that is, return dim(domain(f)) - dim(codomain(f)).

Examples

julia> P2 = abstract_projective_space(2)
AbstractVariety of dim 2

julia> P5 = abstract_projective_space(5, symbol = "H")
AbstractVariety of dim 5

julia> h = gens(P2)[1]
h

julia> i = map(P2, P5, [2*h])
AbstractVarietyMap from AbstractVariety of dim 2 to AbstractVariety of dim 5

julia> dim(i)
-3

pullback(f::AbstractVarietyMap, y::MPolyDecRingElem)

Return the pullback of y via f.

Examples

julia> P2 = abstract_projective_space(2)
AbstractVariety of dim 2

julia> P5 = abstract_projective_space(5, symbol = "H")
AbstractVariety of dim 5

julia> h = gens(P2)[1]
h

julia> i = map(P2, P5, [2*h])
AbstractVarietyMap from AbstractVariety of dim 2 to AbstractVariety of dim 5

julia> H = gens(P5)[1]
H

julia> pullback(i, H)
2*h

pushforward(f::AbstractVarietyMap, x::MPolyDecRingElem)

Return the pushforward of x via f.

Examples

julia> P2 = abstract_projective_space(2)
AbstractVariety of dim 2

julia> P5 = abstract_projective_space(5, symbol = "H")
AbstractVariety of dim 5

julia> h = gens(P2)[1]
h

julia> i = map(P2, P5, [2*h])
AbstractVarietyMap from AbstractVariety of dim 2 to AbstractVariety of dim 5

julia> pushforward(i, h)
2*H^4

tangent_bundle(f::AbstractVarietyMap)

If domain(f) and codomain(f) are given tangent bundles, return the relative tangent bundle of f.

Examples

julia> P2 = abstract_projective_space(2)
AbstractVariety of dim 2

julia> T = tangent_bundle(P2)
AbstractBundle of rank 2 on AbstractVariety of dim 2

julia> PT = abstract_projective_bundle(T)
AbstractVariety of dim 3

julia> pr = structure_map(PT)
AbstractVarietyMap from AbstractVariety of dim 3 to AbstractVariety of dim 2

julia> PBT = pullback(pr, T)
AbstractBundle of rank 2 on AbstractVariety of dim 3

julia> tangent_bundle(PT) - PBT == tangent_bundle(pr)
true

julia> PBT*OO(PT, 1) - OO(PT) == tangent_bundle(pr) # relative Euler sequence
true

cotangent_bundle(f::AbstractVarietyMap)

If domain(f) and codomain(f) are given tangent bundles, return the relative cotangent bundle of f.

todd_class(f::AbstractVarietyMap)

Return the Todd class of the relative tangent bundle of f.

compose(f::AbstractVarietyMap, g::AbstractVarietyMap)

Given abstract variety maps f : X $\to$ Y and g : Y $\to$ Z, say, return their composition.