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

About 30 years old · traced to

Let GG act on XX with finite reduced stabilizers. Write

τXG:K0′G(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 ϵ∈K0G(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

α∈ker⁡ ⁣(τXG:K0′G(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 ϵ∈K0G(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.

References

Primary source

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

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