CurrentModule = Oscar
DocTestSetup = Oscar.doctestsetup()
Recall that our focus in this chapter is on abstract intersection theory: We discuss computations which manipulate collections of data referred to as abstract varieties, and we interprete the results as applying to all (smooth projective complex) varieties sharing the data. The tools presented in this section allow for more efficient computations in the case of varieties with a (split) torus action whose fixed point set is finite. They are based on localization and a version of Bott's formula which is formulated in the language of equivariant intersection theory. See Dan14 and the references cited there.
Using Bott's formula in enumerative geometry goes back to ES02. We quote from that paper:
Many parameter spaces carry natural actions of algebraic tori, in particular those coming from projective enumerative problems. In 1967, Bott gave a residue formula that allows one to express the degree of certain zero-cycles on a smooth complete variety with an action of an algebraic torus in terms of local contributions supported on the components of the fixpoint set. These components tend to have much simpler structure than the whole space; indeed, in many interesting cases, including all the examples of the present paper, the fixpoints are actually isolated.
We represent an abstract variety with a torus action by specifying its dimension together with the fixed points of the action and, possibly, further data.
!!! note In order to work with a version of Bott's formula for orbifolds, it is allowed to specify multiplicities at the fixed points. See the section on Kontsevich moduli spaces.
An abstract equivariant vector bundle under a torus action is represented by its rank and its base variety, together with its localizations at the fixed points.
!!! note Recall that an equivariant vector bundle over a point is a representation of the group under consideration (in our case, a torus).
For our purposes here, we offer the type TnRep.
tn_representation(w::Vector{<:IntegerUnion})
dual(F::TnRep)
The OSCAR type for abstract varieties with a torus action is TnVariety.
tn_variety(n::Int, points::Vector{Pair{P, Int}}) where P
tn_grassmannian(k::Int, n::Int; weights = :int)
tn_flag_variety(dims::Int...; weights = :int)
dim(X::TnVariety)
fixed_points(X::TnVariety)
tangent_bundle(X::TnVariety)
tautological_bundles(X::TnVariety)
As for the type AbstractVariety, we have the methods trivial_line_bundle(X::TnVariety) (alternatively, OO(X::TnVariety)),
cotangent_bundle(X::TnVariety), and euler_number(X::TnVariety). Morever, if X is of type TnVariety, entering total_chern_class(X)
returns the total Chern class of the tangent bundle of X. Similarly for entering chern_class(X, k).
The OSCAR type for an abstract equivariant vector bundle under a torus action is TnBundle.
tn_bundle(X::TnVariety, r::Int, f::Function)
If F is of type TnBundle, then rank(F) and parent(F) return the rank and the
underlying variety of F, respectively. Moreover, we have:
localization(F::TnBundle)
dual(F::TnBundle)
In contrast to the varieties of type AbstractVariety, there are no associated Chow rings for the varieties of type TnVariety.
In order to work with polynomial expressions in the Chern classes of an abstract equivariant vector bundle, Oscar
internally creates an appropriate polynomial ring. We illustrate this in the examples below.
To work with with polynomial expressions in Chern classes, we offer the type TnBundleChern.
chern_class(F::TnBundle, f::RingElem)
total_chern_class(F::TnBundle)
tn_bundle(c::TnBundleChern)
polynomial(c::TnBundleChern)
The usual arithmetic operations are available.
julia> G = tn_grassmannian(1, 3);
julia> T = tangent_bundle(G)
TnBundle of rank 2 on TnVariety of dim 2 with 3 fixed points
julia> c1 = chern_class(T, 1)
Chern class c[1] of TnBundle of rank 2 on TnVariety of dim 2 with 3 fixed points
julia> c2 = chern_class(T, 2)
Chern class c[2] of TnBundle of rank 2 on TnVariety of dim 2 with 3 fixed points
julia> c = c1^2-3*c2
Chern class c[1]^2 - 3*c[2] of TnBundle of rank 2 on TnVariety of dim 2 with 3 fixed points
julia> typeof(c)
TnBundleChern
integral(c::TnBundleChern)
linear_subspaces_on_hypersurface(k::Int, d::Int; bott::Bool = true)