Let (a_1,…,a_n) be a partition of 2g-2 with non-negative parts. For sufficiently large r, let W_g; a_1, …, a_n(r)=r^g-1W^r_g,n(a_1, …, a_n) be the resulting polynomial in r, and let overlineH_g(a_1, …, a_n) denote the closure of the locus of holomorphic differentials with zero orders a_1,…,a_n. Holomorphic differential locus conjecture. (-1)^gW_g; a_1, …, a_n(0)=[overlineH_g(a_1, …, a_n)]∈ H^2(g-1)(overlineM_g,n). This predicts that the class of the compactified holomorphic-differential locus is obtained from the constant term of the polynomial extension of Witten's r-spin class. The claim has been proved by A. Sauvaget, so it is no longer open.

Holomorphic differential locus conjecture

About 10 years old · traced to 2016

Let $(a_1,\ldots,a_n)$ be a partition of $2g-2$ with non-negative parts. For sufficiently large $r$ , let $\mathcal{W}_{g; a_1, \dots, a_n}(r)=r^{g-1}W^r_{g,n}(a_1, \dots, a_n)$ be the resulting polynomial in $r$ , and let $\overline{\mathcal{H}}_g(a_1, \dots, a_n)$ denote the closure of the locus of holomorphic differentials with zero orders $a_1,\ldots,a_n$ . Holomorphic differential locus conjecture.

(-1)^g\mathcal{W}_{g; a_1, \dots, a_n}(0)=[\overline{\mathcal{H}}_g(a_1, \dots, a_n)]\in H^{2(g-1)}(\overline{\mathcal{M}}_{g,n}).

This predicts that the class of the compactified holomorphic-differential locus is obtained from the constant term of the polynomial extension of Witten's $r$ -spin class. The claim has been proved by A. Sauvaget, so it is no longer open.

References

Primary source

R. Pandharipande, A. Pixton and D. Zvonkine, “Tautological relations via r-spin structures”, arXiv:1607.00978 (2020).

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A result attributed to A. Sauvaget says the conjecture is solved, but this scan provides no independent verification.

The conjecture identifies the compactified locus of holomorphic differentials with a class extracted from the polynomial extension of Witten’s $r$ -spin class. The retrieved catalogue records the claim as proved by A. Sauvaget.

Sauvaget proof (July 2016)

The source states that A. Sauvaget proved the conjectural tautological-class formula. Under this report’s verification standard, the result remains an unverified claimed solution because no independent proof assessment was retrieved.

Current status (as of September 2026): The conjecture is reported as proved by A. Sauvaget, but the retrieved evidence does not independently verify the proof.

Sources

No solutions have been posted yet.