For every integer d≥ 1 and every real number varepsilon>0, the class of d-dimensional varepsilon-log-canonical Fano varieties over a field, equivalently varieties X with dim X=d, varepsilon-lc singularities, and ample anticanonical divisor -K_X, is bounded. In the surface case considered by Bhutani and Das, for every varepsilon>0 the class of geometrically integral varepsilon-klt del Pezzo surfaces over arbitrary fields is bounded.

Borisov–Alexeev–Borisov conjecture

Alexeev and the Borisov brothers proposed boundedness for Fano varieties with fixed dimension and singularity threshold. Birkar later proved the conjecture in characteristic zero in all dimensions.

Known results

Birkar (2016) proved characteristic-zero boundedness for fixed dimension and fixed singularity threshold, including pairs with nef and big anticanonical divisor.
The weak three-dimensional form, bounding anticanonical volume in terms of $\varepsilon$ , was established earlier; later work gives sharper quantitative bounds.

September 2026 positive-characteristic surface claim

Bhutani and Das claim that geometrically integral $\varepsilon$ -klt del Pezzo surfaces are bounded in all positive characteristics. Their preprint proves the stated uniform irregularity bound $q(X)\le 1+1/\varepsilon$ in characteristics $2$ , $3$ , and $5$ , completing the remaining surface cases; the claim is unrefereed and unverified.

Current status (as of September 2026): Birkar's characteristic-zero theorem is settled, while the claimed arbitrary-field surface completion awaits verification and these sources do not establish higher-dimensional positive-characteristic cases.

Borisov–Alexeev–Borisov conjecture

Primary source

Additional references

Progress summary

Solutions 0