Vistoli's conjecture on the kernel of equivariant Riemann–Roch

From papers

Let GG act on XX with finite reduced stabilizers. Write

τXG:K0G(X)A([X/G])Q\tau_X^G:K_0^{'G}(X)\longrightarrow A_*([X/G])_{\mathbf Q}

for the equivariant Riemann–Roch map, and let α\alpha lie in its kernel. An element ϵK0G(X)\epsilon\in K_0^G(X) has every non-zero rank if it is represented by a complex of locally free sheaves whose homology is non-zero at the generic point of every subvariety.

Vistoli's conjecture. If

αker ⁣(τXG:K0G(X)A([X/G])Q),\alpha\in\ker\!\left(\tau_X^G:K_0^{'G}(X)\longrightarrow A_*([X/G])_{\mathbf Q}\right),

then there exists an element ϵK0G(X)\epsilon\in K_0^G(X) with every non-zero rank such that

ϵα=0.\epsilon\alpha=0.

The conjecture describes the kernel of the equivariant Riemann–Roch map when the quotient has finite reduced stabilizers. The supplied source does not state whether it has been resolved.

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Sources & referencesView supporting material

Primary source

Dan Edidin and William Graham, “Equivariant intersection theory”, arXiv:alg-geom/9603008 (1996).

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