All Outputs (7)

On K-moduli of quartic threefolds (2025)
Journal Article
Abban, H., Cheltsov, I., Kasprzyk, A., Liu, Y., & Petracci, A. (2025). On K-moduli of quartic threefolds. Algebraic Geometry, 12(3), 382-417. https://doi.org/10.14231/AG-2025-011

The family of smooth Fano 3-folds with Picard rank 1 and anticanonical volume 4 consists of quartic 3-folds and of double covers of the 3-dimensional quadric branched along an octic surface. They can all be parametrised as complete intersections of a... Read More about On K-moduli of quartic threefolds.

One-dimensional components in the K-moduli of smooth Fano 3-folds (2024)
Journal Article
Abban, H., Cheltsov, I., Denisova, E., Etxabarri Alberdi, E., Kaloghiros, A.-S., Jiao, D., Martinez Garcia, J., & Papazachariou, T. (2025). One-dimensional components in the K-moduli of smooth Fano 3-folds. Journal of Algebraic Geometry, 34(3), 489-534. https://doi.org/10.1090/jag/839

By identifying K-polystable limits in 4 specific deformations families of smooth Fano 3-folds, we complete the classification of one-dimensional components in the K-moduli space of smoothable Fano 3-folds.

Double Veronese cones with 28 nodes (2024)
Journal Article
Abban, H., Cheltsov, I., Park, J., & Shramov, C. (2024). Double Veronese cones with 28 nodes. L’Enseignement mathématique, 71(1/2), 1-70. https://doi.org/10.4171/lem/1067

We study nodal del Pezzo 3-folds of degree 1 (also known as double Veronese cones) with 28 singularities, which is the maximal possible number of singularities for such varieties. We show that they are in one-to-one correspondence with smooth plane q... Read More about Double Veronese cones with 28 nodes.

Seshadri constants and K-stability of Fano manifolds (2023)
Journal Article
Abban, H., & Zhuang, Z. (2023). Seshadri constants and K-stability of Fano manifolds. Duke Mathematical Journal, 172(6), 1109-1144. https://doi.org/10.1215/00127094-2022-0026

We give a lower bound of the ı-invariants of ample line bundles in terms of Seshadri constants. As applications, we prove the uniform K-stability of infinitely many families of Fano hypersurfaces of arbitrarily large index, as well as the uniform K-s... Read More about Seshadri constants and K-stability of Fano manifolds.

Recent Developments in Algebraic Geometry: To Miles Reid for his 70th Birthday (2022)
Book
Abban, H., Brown, G., Kasprzyk, A., & Mori, S. (Eds.). (2022). Recent Developments in Algebraic Geometry: To Miles Reid for his 70th Birthday. Cambridge University Press (CUP)

K-stability of Fano varieties via admissible flags (2022)
Journal Article
Abban, H., & Zhuang, Z. (2022). K-stability of Fano varieties via admissible flags. Forum of Mathematics, Pi, 10, Article e15. https://doi.org/10.1017/fmp.2022.11

We develop a general approach to prove K-stability of Fano varieties. The new theory is used to (a) prove the existence of Kähler-Einstein metrics on all smooth Fano hypersurfaces of Fano index two, (b) compute the stability thresholds for hypersurfa... Read More about K-stability of Fano varieties via admissible flags.

Stability of fibrations over one-dimensional bases (2022)
Journal Article
Abban, H., Fedorchuk, M., & Krylov, I. (2022). Stability of fibrations over one-dimensional bases. Duke Mathematical Journal, 171(12), 2461-2518. https://doi.org/10.1215/00127094-2022-0025

We introduce and study a new notion of stability for varieties fibered over curves, motivated by Kollár’s stability for homogeneous polynomials with integral coefficients. We develop tools to study geometric properties of stable birational models of... Read More about Stability of fibrations over one-dimensional bases.