GW/DT4 conjecture for the local projective plane

Let $X=\operatorname{Tot}_{\mathbb{P}^2}\!\left(\mathcal{O}_{\mathbb{P}^2}(-1)\oplus\mathcal{O}_{\mathbb{P}^2}(-2)\right)$ . For every effective curve class $\beta\in H_2(\mathbb{P}^2,\mathbb{Z})$ , the $\mathrm{DT}_4$ invariant of one-dimensional stable sheaves on $X$ in class $\beta$ , with a canonical orientation, should agree with the Gromov–Witten/Gopakumar–Vafa-type expression proposed for the local projective plane: $\mathrm{DT}_4(\beta;X)=\mathrm{GW/GV}(\beta;X)$ .

References

Primary source

arXiv

Additional references

A proof of GW/DT4 conjecture on local projective plane — arXiv — Yalong Cao

Posted 1 day ago · edited 1 day ago

Progress summary

Refreshed 1 day ago

Claimed solved

An unrefereed preprint claims to settle the conjecture for one important geometric example, but the claim has not been independently verified.

The problem concerns the proposed $\mathrm{GW}/\mathrm{DT}_4$ correspondence for the local projective plane, a Calabi–Yau fourfold test case. The reported result gives a closed calculation with canonical orientations and compares it with the proposed Gopakumar–Vafa-type invariants.

October 2026 preprint

Yalong Cao claims a complete proof for the local projective plane, deriving a closed $\mathrm{DT}_4$ formula with canonical orientations and matching it to the proposed invariants. This would establish the conjectural correspondence in the stated geometry, but the preprint is unrefereed and the claim is therefore unverified.

Current status (as of October 2026): The conjecture is claimed solved for the local projective plane by an unrefereed preprint, but no verified resolution is recorded.

Sources

No solutions have been posted yet.