Today I finally understood what Bernd would like me to prove, so I thought I'd write it down here.
First we must understand the localization statement for a scheme \(X\) equipped with the action of a torus \(T\). Denote by \(\hat T = \operatorname{Hom}_{\mathsf{Grp}}(T,\mathbb C^\times)\) the character group of \(T\) and \(S = \operatorname{Sym}(\hat T)\) the symmetric algebra over \(\math Z\) of \(\hat T\) as an Abelian group. The equivariant Chow group of a point identifies to \(S\), and more generally, \(A_*^T(X)\) is a \(S\)-module. The statement of localization for \(X\) then says that the inclusion map \(i:X^T\to X\) of the fixed points into \(X\) induces an isomorphism
\[ i_*:A^T_*(X^T) \to A^T_*(X) \]after inverting all nonzero elements of \(S\). See Corollary 2 of ["Equivariant Chow Groups For Torus Actions", M. Brion] for this statement and any of the Edin Graham papers for the definitions of equivariant Chow groups.
This statement makes no assumptions on \(X\); in particular, we don't require \(X\) to be nonsingular. If we do have nonsingularity however we can say more. In this case we have a ring structure on \(A^*_T(X)\), the equivariant Chow ring, given by intersection product. It's more or less the collection of the following facts [Corollary 3.2.1, "Equivariant Chow Groups For Torus Actions, M. Brion].
Localization. Let \(X\) be a nonsingular filtrable \(T\)-variety, i.e. a nonsingular variety with a \(T\)-action which is "filterable". Then
The inclusion map \(i:X^T\to X\) induces an injective \(S\)-algebra homomorphism \(i^*:A^*_T(X)_{\mathbb Q} \to A^*_T(X^T)_{\mathbb Q}\) which is surjective over the quotient field of \(S\).
The \(S_{\mathbb Q}\)-module \(A_*^T(X)_{\mathbb Q}\) is free. If moreover the \(\mathbb Z\)-module \(A_*(X^T)\) is free, then the \(S\)-module \(A_*^T(X)\) is free.
If \(X^T\) consists of finitely many points \(x_1,...,x_m\), then, for any generic one-paramter subgroup \(\lambda\), the \(S\)-module \(A_*^T(X)\) is freely generated by the classes of the closures of strata \(X_+(x_i,\lambda)\) for \(1\leq i \leq m\).
There's stuff I'm not defining here – "filterable" and "\(X_+(x_i, \lambda)\)" for instance – but the important takeaways are that
there is always a localization isomorphism for a variety \(X\) with a toric action \(T\)
there is a nice localization formula when \(X\) is nonsingular.
The current project is to investigate the existence of a localization *formula in the case of toric singularities. Here are some questions/comments that resulted from today:*
Which toric varieties are complete intersections? Edin Graham have a localization formula in the case that \(X\hookright M = \mathbb P^N\) is a complete intersection. This question is likely going to be gross – "toric" and "complete intersection" don't seem to have much to do with one another.
Starting from scratch, understand the localization theorem in the Edin Graham paper and in the M. Brion paper. Look at the toric variety corresponding to the cone over a square (this is not simplicial, simplicial toric varieties have extra stuff you can say about them).
Try to start thinking of examples, beginning with a singular surface perhaps, and then computing a localization formula for it.
Read some of the examples in section 1.6 of [Convex Bodies in Algebraic Geometry, Oda]. Need to review stuff about toric singularities and refinements of cones etc.