Gökova Geometry Topology Institute

The research session will bring together the following experts on Algebraic Geometry.

During our visit, we plan to obtain comprehensive results on K-stability of singular hyperelliptic Fano 3-folds and smooth Mukai 4-folds. In particular, we aim to classify all K-polystable double covers of three-dimensional projective space branched along singular sextic surfaces, and we plan to prove K-polystability of all smooth Fano-Mukai 4-folds of genus 10 that have reductive automorphism group.

Smooth Fano 3-folds have been classified by Iskovskikh, Mori and Mukai into 105 deformation families. The description of these families are available online at the following web page: https://www.fanography.info. In 2019, Cheltsov organized a collaborative research project on K-stability of smooth Fano 3-folds. This project resulted in many publications including 450 pages long book “The Calabi problem for Fano threefolds” by Carolina Araujo, Ana-Maria Castravet, Ivan Cheltsov, Kento Fujita, Anne-Sophie Kaloghiros, Jesus Martinez-Garcia, Constantin Shramov, Hendrik Suess, Nivedita Viswanathan, which has been published in 2023 at Cambridge University Press (London Mathematical Society Lecture Notes Series, volume 485).

Now it is time to study K-stability of singular Fano 3-folds and smooth Fano 4-folds. Despite efforts of many mathematicians, these geometric objects are not yet classified. However, we know classification of singular Fano 3-folds with Gorenstein canonical singularities such that their anticanonical linear systems give double covers (due to Cheltsov, Przyjalkowski, Shramov), which are known as hyperelliptic Fano 3-folds. Similarly, we know classification of smooth Fano 4-folds whose anticanonical divisors are divisible by 2 in the Picard group (due to Mukai and Wisniewski), which are known as Mukai 4-folds (or Fano-Mukai 4-folds). The list of hyperelliptic Fano 3-folds consists of 47 deformation families (only 6 families contain smooth members), and the list of smooth Mukai 4-folds consists of 29 deformation families. At the moment, we know very few results about K-stability of these 3-folds and 4-folds, e.g. all smooth hyperelliptic Fano 3-folds are known to be K-polystable (due to Abban, Cheltsov, Denisova, Dervan, Fujita, Park, Tian, Zhuang). We plan to fill this gap during our research session at Gokova Geometry Topology Institute.

To start with, we plan to find all K-polystable double covers of three-dimensional projective space branched along singular reduced (but possibly reducible) sextic surfaces - these are hyperelliptic Fano 3-folds of anticanonical degree 2 (in the smooth case, their K-stability has been proven by Cheltsov and Park). Then we plan to classify K-polystable smooth Mukai 4-folds of genus 10. These Mukai 4-folds have been studied by Prokhorov and Zaidenberg in a series of papers, and their geometry is well-understood. Up to isomorphisms, they form a one-dimensional family, and one 4-fold in this family has a non-reductive automorphism group, so it is not K-polystable by the Matsushima obstruction. We expect that the remaining smooth Mukai 4-folds of genus 10 are K-polystable and we plan to prove this during our visit to Gokova. If time permits, we plan to study K-stability of other singular hyperelliptic Fano 3-folds and smooth Mukai 4-folds.

For further questions about this event please contact : Ivan Cheltsov