Berndtsson–Darvas–Zhang uniqueness conjecture
For fixed singular or big-cohomology data for which twisted Kähler–Einstein currents are defined, if and are two twisted Kähler–Einstein currents satisfying the same twisted Kähler–Einstein equation, then . Equivalently, the relevant twisted Kähler–Einstein current is unique whenever it exists.
References
Primary source
Additional references
- A solution to Berndtsson's problem and uniqueness of twisted KE currents — arXiv — Yinji Li, Haoyuan Sun, Zhiwei Wang, Xiangyu Zhou
Progress summary
A September 2026 preprint claims to prove uniqueness of these canonical geometric objects, but the claim has not yet been independently verified.
The conjecture asks whether twisted Kähler–Einstein currents in singular or big cohomology settings are unique, generalizing the Bando–Mabuchi theorem. The associated uniqueness problem was posed by Bo Berndtsson.
Known results
- Berndtsson (2011) proved classical and twisted uniqueness under additional positivity, regularity, and integrability assumptions.
- Berndtsson (2013) established uniqueness for twisting currents given by integration over a klt divisor.
- Darvas–Zhang (2022; revised 2024) proved existence under divisorial stability, including solutions when , but explicitly left uniqueness open.
September 2026 claimed proof
On September 16, 2026, a preprint by Yinji Li, Haoyuan Sun, Zhiwei Wang, and Xiangyu Zhou claimed a Bando–Mabuchi-type uniqueness theorem resolving Berndtsson’s problem. This is an unrefereed solution claim and has no independent verification in the retrieved sources.
Current status (as of September 2026): A general uniqueness theorem is claimed in the Li–Sun–Wang–Zhou preprint, while the claim remains unverified; earlier special cases and existence results are established.
Solutions 0
No solutions have been posted yet.