Polishchuk–Van den Bergh Conjecture A

For every effective action of a finite group $G$ on a smooth projective complex curve $X$ , there should exist an ordering of the conjugacy classes $[g]$ of $G$ and a semiorthogonal decomposition $D^b_G(X)=\langle \Phi_{[g]}(D^b(X^g/C_G(g)))\rangle_{[g]}$ , where $X^g$ is the fixed-point locus of $g$ , $C_G(g)$ is the centralizer of $g$ , and each functor $\Phi_{[g]}:D^b(X^g/C_G(g))\hookrightarrow D^b_G(X)$ is an exact $\mathbb{C}$ -linear fully faithful embedding.

References

Primary source

arXiv

Additional references

Counterexamples to the Polishchuk--Van den Bergh conjecture on curves — arXiv — Shengyong Pan

Posted 7 hours ago · edited 7 hours ago

Progress summary

Refreshed 7 hours ago

Claimed progress

A new unrefereed preprint claims the conjecture fails for certain curve symmetries, but this has not been independently checked.

Polishchuk and Van den Bergh proposed a categorical decomposition for equivariant derived categories of finite reflection groups. Their paper was first submitted in 2015, revised in 2017, and published in 2019; version 2 added effectiveness of the group action to Conjecture A.

Known results

For several groups, including Weyl groups of types $A$ , $B$ , $G_2$ , and $F_4$ , and groups $G(m,1,n)$ , the paper constructs decompositions indexed by conjugacy classes; this does not prove Conjecture A in full (Polishchuk–Van den Bergh, 2015–2017).

October 2026 claimed counterexamples

Shengyong Pan claims that explicit $S_3$ actions on hyperelliptic curves violate the conjectured fully faithful embeddings and proposes a connected-component replacement criterion. The preprint is unrefereed, so the counterexamples and correction remain unverified.

Current status (as of October 2026): The original constructions and hypotheses are established, while Pan's claimed counterexamples and replacement criterion are not independently verified, so the general status of Conjecture A remains open.

Sources

No solutions have been posted yet.