MathDB

Yang–Zheng quasi-negative curvature conjecture

Let XX be a compact complex manifold. If XX admits a Hermitian metric hh whose Chern holomorphic sectional curvature satisfies Hh(v)≤0H_h(v)\le 0 for every nonzero v∈Tx1,0Xv\in T^{1,0}_xX and is strictly negative at some point and in some nonzero direction, then the canonical bundle KXK_X is ample.

References

Primary source

arXiv

Additional references

complex-geometrykhler-geometry
Posted · edited

Progress summary

Refreshed
Claimed progress

A new preprint claims the conjecture in the compact Kähler case, while the version for all compact complex manifolds remains open.

The conjecture asserts that a compact complex manifold with a Hermitian metric of quasi-negative holomorphic sectional curvature has ample canonical bundle. The broader statement is attributed to Yang and Zheng; the retrieved material does not establish a complete resolution.

Known results

  • Diverio–Trapani and Wu–Yau, 2016: for a Kähler metric with quasi-negative holomorphic sectional curvature, the canonical bundle is ample.
  • Yang–Zheng: ampleness under the stronger assumption of quasi-negative real bisectional curvature.
  • Broder–Stanfield: for a pluriclosed Hermitian metric with nonpositive holomorphic sectional curvature, the canonical bundle is nef, and it is ample under strict negativity.

October 2026 Kähler-case advance

Xueyuan Wan's preprint claims the implication for compact Kähler manifolds even when the curvature-producing Hermitian metric is not Kähler. The full assertion for arbitrary compact complex manifolds remains open; this claimed advance is unverified.

Current status (as of October 2026): The compact Kähler case is claimed in a recent preprint, but the conjecture for arbitrary compact complex manifolds remains open and the new claim is unverified.

Sources

Solutions 0

No solutions have been posted yet.