Every projectively induced Kähler–Einstein metric on a smooth compact toric manifold is, up to automorphisms, a product of Fubini–Study metrics on projective spaces with matched multiples; equivalently, its projective immersion is given by a complete Veronese–Segre system up to automorphisms. In particular, the underlying manifold is biholomorphic to a product CP^n_1×…×CP^n_r.

Homogeneity conjecture for projectively induced toric Kähler–Einstein metrics

The conjecture asserts that projectively induced toric Kähler–Einstein metrics have the rigid product-homogeneous form predicted by the associated classification. The newer claim concerns the compact toric setting, not the broader homogeneity conjecture.

Known results

The conjecture was known in complex dimension $n=2$ .
The toric case was known through dimension $n=4$ .
A 2024 paper proved the stated conjecture for $n\leq 6$ .
Salis (2016) proved related rotation-invariant cases under codimension at most $3$ .

September 16, 2026 claimed resolution

Shaosai Huang’s paper Generalized Einstein Laurent polynomials, Toric Kähler-Einstein Rigidity, and Finite Exponential Families claims a proof of the compact toric homogeneity conjecture and related Manno–Salis conjectures, together with finite-support cases of a question of Casalis. The claim is not independently verified in the retrieved material.

Current status (as of September 2026): the compact toric conjecture is claimed solved by Huang’s paper but remains unverified; broader homogeneity questions remain open or outside the paper’s scope.

Homogeneity conjecture for projectively induced toric Kähler–Einstein metrics

Primary source

Additional references

Progress summary

Solutions 0