Abstract variety maps
An abstract variety map $f\colon X \to Y$ encodes a morphism between abstract varieties at the level of intersection theory. It is determined by:
- the pullback $f^*\colon \mathrm{N}^*(Y)_{\mathbb Q}\to \mathrm{N}^*(X)_{\mathbb Q}$, a ring homomorphism on Chow rings;
- optionally, a pushforward $f_*\colon \mathrm{N}^*(X)_{\mathbb Q}\to \mathrm{N}^*(Y)_{\mathbb Q}$, a group homomorphism satisfying the projection formula.
When the pushforward is not given explicitly, it can sometimes be computed automatically via the projection formula, provided enough information about the Chow ring of $Y$ is available (see the documentation of map for details).
Abstract variety maps also carry a relative tangent bundle $\mathrm{T}_f$, which satisfies $\mathrm{T}_X = f^* \mathrm{T}_Y \oplus \mathrm{T}_f$, and may carry a relative polarization $\mathcal{O}_f(1)$.
Pullback and pushforward semantics
The pullback $f^*$ is a ring homomorphism: it preserves products and sends $1$ to $1$. It is always available once a map is constructed.
The pushforward $f_*$ is only a group homomorphism (it does not preserve products in general), but it satisfies the projection formula:
\[f_*(f^*(y) \cdot x) = y \cdot f_*(x) \quad \text{for all } x \in \mathrm{N}^*(X),\; y \in \mathrm{N}^*(Y).\]
When the pushforward is not specified explicitly, OSCAR attempts to compute it from the pullback using the projection formula together with the intersection pairing on $Y$. This succeeds automatically when:
- Y is a point or a curve;
- all algebraic classes in $\mathrm{N}^*(Y)_{\mathbb Q}$ are known (e.g., projective spaces, Grassmannians, complete intersections);
- the flag
:algis set on $Y$.
In other cases, a warning is issued and the result may be incorrect.
julia> P2 = abstract_projective_space(2);
julia> P5 = abstract_projective_space(5, symbol = "H");
julia> h = gens(P2)[1]
h
julia> i = map(P2, P5, [2*h]); # Veronese embedding
julia> pushforward(i, h) # pushforward works via projection formula
2*H^4
julia> pushforward(i, P2(1)) # pushforward of the fundamental class
4*H^3
Inclusions and extend_inclusion
For an inclusion $i\colon X \hookrightarrow Y$, where the class $[X]$ is not already representable in $\mathrm{N}^*(Y)_{\mathbb Q}$, the pushforward $i_*$ cannot be expressed in the existing Chow ring of $Y$. Setting inclusion = true in the map constructor calls extend_inclusion to create a modified variety $Y^+$ with extra generators added to the Chow ring so that $i_*$ becomes well-defined. The structure map $Y^+ \to Y$ gives access to the enlarged ring while preserving the original ring via pullback.
Types
The OSCAR type for abstract variety maps is AbstractVarietyMap.
Constructors
map — Function
map(X::AbstractVariety, Y::AbstractVariety,
f_pullback::Vector, f_pushforward = nothing;
inclusion::Bool = false, symbol::String = "x")Return an abstract variety map $f:X \rightarrow Y$ by specifying the pullbacks of the generators of the Chow ring $\mathrm{N}^*(Y)_{\mathbb Q}.$ If needed, also specify the pushforward map $\mathrm{N}^*(X)_{\mathbb Q} \rightarrow \mathrm{N}^*(Y)_{\mathbb Q}.$
The pullback is relatively easy to describe since it is functorial, while the pushforward is usually more complicated. In some cases, the pushforward of an element $x \in \mathrm{N}^*(X)_{\mathbb Q}$ can be automatically computed via pullback. Specifically, the projection formula tells us that
\[f_\ast(f^\ast(y)\cdot x) = f_\ast(x)\cdot y \;\text{ for all }\; x\in \mathrm{N}^*(X)_{\mathbb Q}, y\in \mathrm{N}^*(Y)_{\mathbb Q}.\]
Since we are working with numerical equivalence, to determine $f_\ast(x)$, it suffices to know all its intersection numbers. These can be readily computed via pullback, provided that all classes in $\mathrm{N}^*(Y)_{\mathbb Q}$ are known (or at least those classes having non-zero intersection numbers with $f_\ast(x)$). This is the case in the following situations:
- When $Y$ is a point or a curve;
- when all classes in $\mathrm{N}^*(Y)_{\mathbb Q}$ are known;
- when
:algis passed as the fourth argument. This can be done when we are certain that the computed pushforward is correct, even though not all classes in $\mathrm{N}^*(X)_{\mathbb Q}$ are known.
In the other cases, if no pushforward map has been specified, a warning will be given when trying to do pushforward.
In the case of an inclusion $X \hookrightarrow Y$ where the class of $X$ in $\mathrm{N}^*(Y)_{\mathbb Q}$ is not already representable in $\mathrm{N}^*(Y)_{\mathbb Q}$, use the argument inclusion = true. Then, a modified version $Y^+$ of $Y$ will be created, with extra classes added to the Chow ring so that one can pushforward all classes on $X$. In this case, the string symbol specifies how the extra generators of the extended Chow ring $\mathrm{N}^*(Y^+)_{\mathbb Q}$ are printed. See the subsection Example: Cubic fourfolds of the documentation for an example.
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
julia> pushforward(Se, pullback(Se, h)*(k+l)) == h*pushforward(Se, k+l)
true
This function is part of the experimental code in Oscar. Please read here for more details.
extend_inclusion — Method
extend_inclusion(i::AbstractVarietyMap; symbol::String = "x")Given an inclusion $i\colon X \hookrightarrow Y$ where the class of $X$ in $\mathrm{N}^*(Y)_{\mathbb Q}$ is not already representable in $\mathrm{N}^*(Y)_{\mathbb Q}$, create a modified version $Y^+$ of $Y$ with extra classes added to the Chow ring so that one can pushforward all classes on $X$.
Return the resulting morphism $j\colon X \to Y^+$.
The string symbol specifies how the extra generators in the extended Chow ring $\mathrm{N}^*(X^+)_{\mathbb Q}$ are printed.
See the subsection Example: Cubic fourfolds of the documentation for an example.
This function is part of the experimental code in Oscar. Please read here for more details.
identity_map — Method
identity_map(X::AbstractVariety)Return the identity map on X.
This function is part of the experimental code in Oscar. Please read here for more details.
This function is part of the experimental code in Oscar. Please read here for more details.
Underlying data of an abstract variety map
An abstract variety map is made up from (a selection of) the data discussed here:
dim — Method
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
This function is part of the experimental code in Oscar. Please read here for more details.
pullback — Method
pullback(f::AbstractVarietyMap, y::MPolyDecRingOrQuoElem)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
This function is part of the experimental code in Oscar. Please read here for more details.
pushforward — Method
pushforward(f::AbstractVarietyMap, x::MPolyDecRingOrQuoElem)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^4This function is part of the experimental code in Oscar. Please read here for more details.
tangent_bundle — Method
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 = 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
This function is part of the experimental code in Oscar. Please read here for more details.
Further data associated to an abstract variety map
cotangent_bundle — Method
cotangent_bundle(f::AbstractVarietyMap)If domain(f) and codomain(f) are given tangent bundles, return the relative cotangent bundle of f.
This function is part of the experimental code in Oscar. Please read here for more details.
todd_class — Method
todd_class(f::AbstractVarietyMap)Return the Todd class of the relative tangent bundle of f.
Examples
The fibre variable z is the relative polarization of the projective bundle:
julia> P2 = abstract_projective_space(2);
julia> PF = projective_bundle(tautological_bundles(P2)[2]);
julia> z = polarization(PF) # relative O(1) class on the fibre
z
julia> todd_class(structure_map(PF))
z - 1//4*h^2 + 1//2*h + 1
This function is part of the experimental code in Oscar. Please read here for more details.