Torsion-order conjecture of Juteau–Williamson

Let $X$ be a rationally smooth complex affine variety with an algebraic torus action, and let $x\in X$ be an attractive fixed point. Let $p$ be a prime, assume that $X\setminus\{x\}$ is $p$ -smooth and that its integral equivariant cohomology is $p$ -torsion-free. Write the equivariant multiplicity at $x$ in reduced form, and let $n_x$ denote its reduced numerator. Then the conjecture asserts that the order of the total $p$ -primary cohomological torsion associated with $X\setminus\{x\}$ equals the $p$ -part of $n_x$ : $\left|\operatorname{Tor}_p H^\bullet(X\setminus\{x\};\mathbb Z)\right|=\left|n_x\right|_p$ .

References

Primary source

arXiv

Additional references

Equivariant multiplicities and torsion at attractive fixed points — arXiv — Tao Gui, Peter L. Guo, Zhuowei Lin

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed solved

A September 2026 preprint claims to settle the conjecture under explicit smoothness and torsion-freeness assumptions, but its proof has not been independently verified.

The conjecture links equivariant multiplicities at attractive fixed points with the size of integral cohomological torsion. Earlier work established related $p$ -smoothness criteria while recording the torsion-order equality as conjectural.

Known results

Kumar’s criterion modulo $p$ gives, under $p$ -smoothness and torsion-freeness hypotheses, an equivalence between $p$ -smoothness and the reduced numerator being integral and not divisible by $p$ (2012).

September 2026 claimed resolution

Tao Gui, Peter L. Guo, and Zhuowei Lin’s preprint Equivariant multiplicities and torsion at attractive fixed points claims that total $p$ -primary cohomology torsion equals the $p$ -part of the reduced numerator of the equivariant multiplicity. The claim applies under rational smoothness, $p$ -smoothness of the punctured space, and the stated torsion-freeness assumption; it has not been independently verified.

Current status (as of September 2026): The conjectural equality is claimed proved under the stated hypotheses, but independent verification is not recorded; the unrestricted scope remains open.

Sources

No solutions have been posted yet.