The Calabi problem for smooth Fano threefolds
Lead Research Organisation:
Brunel University London
Department Name: Mathematics
Abstract
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People |
ORCID iD |
| Anne-Sophie Kaloghiros (Principal Investigator) |
Publications
Abban H
(2024)
One-dimensional components in the K-moduli of smooth Fano 3-folds
in Journal of Algebraic Geometry
Araujo C
(2023)
The Calabi Problem for Fano Threefolds
Guerreiro T
(2024)
On K-stability of P3$\mathbb {P}^3$ blown up along a (2,3) complete intersection
in Journal of the London Mathematical Society
| Description | The Calabi problem asks which compact complex manifolds can be endowed with a ''special" canonical metric - a Kähler metric that satisfies Einstein's equation. Such metrics can only exist on manifolds which are flat, negatively curved or positively curved. In the first two cases, the answer to the Calabi problem has been known since the 1970s: such a special metric always exist. In the case of positive curvature - the so called Fano varieties - the situation is more subtle. The Yau-Tian-Donaldson conjecture (now a theorem due to Chen, Donaldson and Sun in 2015) states that a Fano manifold admits a Kähler-Einstein metric precisely when it satisfies a sophisticated algebraic condition called K-polystability. Recent results have shown that K-polystable Fano varieties form well behaved moduli spaces, a key property that enables us to understand better how they vary in families. These developments have placed the notion of K-polystability centre stage in current research in differential geometry, geometric analysis and in algebraic geometry. Yet, our understanding of K-polystability remains limited. The research funded by this award has focused on understanding explicit K-polystability of Fano threefolds. The monograph "The Calabi problem for Fano threefolds" is the first comprehensive and systematic approach to the question of K-polystability of Fano threefolds. It is known that there are 105 deformation families of Fano threefolds. For each family, we determine whether the general member of the family is K-polystable (this waspreviously known for 65 families). Further, for 71 out of the 105 deformation families, we determine precisely which members are K-polystable. Other outcomes of the research funded by this award are explicit studies of (components of ) K-moduli spaces of smooth Fano threefolds associated to specific families. |
| Exploitation Route | The outcomes of this funding have established a new research area in Algebraic Geometry studying K-moduli components of smooth Fano threefolds, which has been very active in the past couple of years. |
| Sectors | Other |
| Description | The outcomes of this research have had significant impact in academia and have led to the development of a new research area investigating components of K-moduli spaces of Fano threefolds. This area has been especially active in the past 2 years. |
| First Year Of Impact | 2023 |
| Sector | Education,Other |
| Impact Types | Cultural |
| Description | LMS Emmy Noether Fellowship |
| Amount | £1,800 (GBP) |
| Funding ID | EN-2223-02 |
| Organisation | London Mathematical Society |
| Sector | Academic/University |
| Country | United Kingdom |
| Start | 07/2023 |
| End | 07/2024 |
| Description | Collaboration with Dr Carolina Araujo |
| Organisation | Instituto Nacional de Matemática Pura e Aplicada |
| Country | Brazil |
| Sector | Public |
| PI Contribution | Joint work on modular interpretation of the K-stability of a family of Fano varieties |
| Collaborator Contribution | Joint work on modular interpretation of the K-stability of a family of Fano varieties |
| Impact | Work in progress |
| Start Year | 2023 |
| Description | Collaboration with Dr Kento Fujita, Dr Ivan Cheltsov and Dr Maksym Fedorchuk |
| Organisation | Boston College |
| Country | United States |
| Sector | Academic/University |
| PI Contribution | Joint work on explicit K-stability of Fano threefolds |
| Collaborator Contribution | Joint work on explicit K-stability of Fano threefolds |
| Impact | Preprint to be uploaded to ArXiv on K-moduli of Fano threefolds of rank 4 and anticanonical degree 24 |
| Start Year | 2023 |
| Description | Collaboration with Dr Kento Fujita, Dr Ivan Cheltsov and Dr Maksym Fedorchuk |
| Organisation | Osaka University |
| Country | Japan |
| Sector | Academic/University |
| PI Contribution | Joint work on explicit K-stability of Fano threefolds |
| Collaborator Contribution | Joint work on explicit K-stability of Fano threefolds |
| Impact | Preprint to be uploaded to ArXiv on K-moduli of Fano threefolds of rank 4 and anticanonical degree 24 |
| Start Year | 2023 |
| Description | Collaboration with Dr Kento Fujita, Dr Ivan Cheltsov and Dr Maksym Fedorchuk |
| Organisation | University of Edinburgh |
| Country | United Kingdom |
| Sector | Academic/University |
| PI Contribution | Joint work on explicit K-stability of Fano threefolds |
| Collaborator Contribution | Joint work on explicit K-stability of Fano threefolds |
| Impact | Preprint to be uploaded to ArXiv on K-moduli of Fano threefolds of rank 4 and anticanonical degree 24 |
| Start Year | 2023 |