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Datar–Mete–Song J-equation conjecture

Let XX be a compact Kähler manifold of complex dimension nn, and let α,β\alpha,\beta be Kähler classes on XX. Set μ=n αn−1⋅βαn\mu=\frac{n\,\alpha^{n-1}\cdot\beta}{\alpha^n}. The Datar–Mete–Song conjecture asserts that the pair (α,β)(\alpha,\beta) is semistable if and only if every modification π:Y→X\pi:Y\to X and every big and nef class LL on YY satisfy nLn−1⋅π∗β≥μLnnL^{n-1}\cdot\pi^*\beta\geq\mu L^n. Equivalently, semistability is characterized by ζmin⁡(α,β)=μ\zeta_{\min}(\alpha,\beta)=\mu; in the unstable case, ζmin⁡(α,β)<μ\zeta_{\min}(\alpha,\beta)<\mu.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Minimal-slope formulation

    For a compact Kähler manifold XX of complex dimension nn and Kähler classes α,β\alpha,\beta, with μ=n αn−1⋅βαn\mu=\frac{n\,\alpha^{n-1}\cdot\beta}{\alpha^n}, the pair (α,β)(\alpha,\beta) is semistable if and only if its birational minimal slope satisfies ζmin⁡(α,β)=μ\zeta_{\min}(\alpha,\beta)=\mu; instability is equivalent to ζmin⁡(α,β)<μ\zeta_{\min}(\alpha,\beta)<\mu.

    source: The J-equation at the birational minimal slope

References

Primary source

arXiv

Additional references

complex-geometrykhler-geometrypde
Posted · edited

Progress summary

Refreshed
Claimed solved

A preprint claims to settle the central conjecture, but its broader smoothness conclusions remain incomplete and the work is not peer reviewed.

The Datar–Mete–Song conjecture relates semistability of a pair of Kähler classes to the smallest slope of birational test classes. A new preprint claims the characterization for compact Kähler manifolds, while related weak-solution and regularity questions remain only partly resolved.

Known results

  • Datar and Mete–Song had confirmed the conjecture in complex dimension two.
  • Datar–Mete–Song had already proved the strict inequality in the unstable case.
  • Murakami confirmed the weak JJJJ-equation conjecture in the semistable case using a closed positive current; the unstable weak-solution case remains open.

September 2026 developments

The preprint On the Datar–Mete–Song Minimal Slope Conjecture claims Conjecture 1: semistability is equivalent to every big and nef birational test class satisfying nLn−1⋅π∗β≥μLnnL^{n-1}\cdot\pi^*\beta\geq\mu L^n, equivalently ζmin⁡(α,β)=μ\zeta_{\min}(\alpha,\beta)=\mu; instability gives ζmin⁡(α,β)<μ\zeta_{\min}(\alpha,\beta)<\mu. It also claims toric partial regularity. A separate September preprint claims smoothness only away from the numerical JJ-null locus, not across it.

Current status (as of September 2026): the minimal-slope conjecture is claimed proved by an unrefereed preprint, while the unstable weak-solution case and full smooth regularity remain open.

Sources

Solutions 0

No solutions have been posted yet.