The Calabi Project for Fano threefolds.

Kähler-Einstein Fano 3-folds

In 2023, some of us and many other collaborators, published a book that showed which families of smooth Fano threefolds had at least one smooth Fano variety that admitted a Kähler-Einstein metric. This built on existing work by many previous mathematicians, and specially on the then recently developed Abban-Zhuang technique. The aim of this project is to find all Kähler-Einstein smooth Fano 3-folds. Our working party is checking and rewriting the proofs generated by ChatGPT 6.1 Astra for non-prime smooth Fano 3-folds.

The working party includes: Ivan Cheltsov, Kristin DeVleming, Tiago Duarte Guerreiro, Lena Ji, Anne-Sophie Kaloghiros, Daniel Mallory and Jesus Martinez-Garcia.

Iskovskikh-Mori-Mukai classification

Fanography

What is done?

  • The book: the classification is completed there except for Families 1.9, 1.10, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 2.10, 2.11, 2.12, 2.13, 2.14, 2.15, 2.16, 2.17, 2.18, 2.19, 2.20, 2.21, 2.22, 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8, 3.11, 3.12 and 4.1.
  • Yuchen: K-stability of every smooth member of Family 2.8. (paper)
  • Tiago, Luca and Nivedita: K-stability of every smooth member of Family 2.15.
  • Vanya, Kento, Takashi and Jihun: K-stability of every smooth member of Families 2.18 and 3.4.
  • Joe: the K-polystable members of Family 2.21, with explicit exceptional cases. (paper)
  • Vanya and Jihun: the K-polystable members of Family 2.22.
  • Vanya, Kento, Takashi and Takuzo: K-stability of every smooth member of Family 3.3.
  • Lena: the complete classification of K-polystable smooth members of Family 3.12. (paper)
  • Costya and Grisha: K-stability of every smooth member of Family 4.1.
  • Vanya, Lena and Kento: K-stability of every smooth member of Families 2.1, 2.2, 2.3, 2.4, 2.6 and 2.7; also Family 2.5 when no fibre of the cubic-surface fibration has a Du Val singular point of type D5 or E6. (paper)
  • Tiago, Luca and Nivedita: K-stability of infinitely many smooth members of Family 2.19. (paper)
  • Junyan: K-stability of every smooth member of Family 2.19 and a description of its K-moduli space. (paper)
  • Kento: K-stability of a particular member, and hence of a general member, of Family 3.11. (paper)
  • Vanya, Anne-Sophie, Robert and Junyan: K-stability of every smooth member of Family 3.11 and a classification of its K-semistable degenerations. (paper)
  • Vanya, Oliver, Sione and Antoine: K-stable examples in Family 2.12 and a reduction of possible destabilizing centres to points of the intersection of the two exceptional divisors. (paper)
  • Vanya, DongSeon, Alex, Antonio, Frédéric, Alex, Jihun and Junyan: K-stability criteria for Family 2.16, including every smooth member whose conic-bundle discriminant is smooth. (paper)
  • Anne-Sophie, Yuchen, Andrea and Junyan: every K-semistable degeneration of members of Family 2.16 is the blow-up of a (2,2)-complete intersection in ℙ5 along a conic; the K-moduli stack of Family 2.16 is smooth (Appendix A, Theorem A.1). (paper)
  • Vanya and Piotr: K-stability criteria and infinitely many K-stable examples in Family 2.17. (paper)
  • Grisha and Costya: K-stability in Family 3.2 under an explicit generality condition on the singularities of the cubic fibres containing double conics. (paper)
  • Vanya: K-stability in Family 3.6 when the conic bundle has only reduced fibres. (paper)
  • Lena and Theodoros: an announced K-classification and GIT description of the K-moduli space of Family 3.5; the manuscript is in preparation. (announcement)

AI generated solution (non-prime smooth Fano 3-folds)

Kähler-Einstein Fano 3-folds (PDF version)

Reading group (non-prime smooth Fano 3-folds)

What is left?

  • Families 1.9, 1.10.