Looijenga's conjecture on the smooth Torelli group of Enriques surfaces

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Let YY be an Enriques surface, and let H∗(Y,Z)H^*(Y,\mathbb{Z}) be its integral cohomology equipped with the quadratic form QYQ_Y. Define the smooth Torelli group by

IY:=ker⁡(Mod⁡(Y)→Aut⁡(H∗(Y,Z),QY)).\mathcal{I}_Y:=\ker\bigl(\operatorname{Mod}(Y)\to\operatorname{Aut}(H^*(Y,\mathbb{Z}),Q_Y)\bigr).

Looijenga's conjecture. The smooth Torelli group is trivial:

IY={1}.\mathcal{I}_Y=\{1\}.

The conjecture concerns whether every mapping class of an Enriques surface that acts trivially on integral cohomology is itself trivial. It was suggested by Looijenga; the supplied source does not state whether it has been resolved.

References

Primary source

Sidhanth Raman, “A smooth Birman-Hilden theory for hyperkähler manifolds”, arXiv:2402.17047 (2025).

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