Exceptional del Pezzo hypersurfaces

We compute global log canonical thresholds of a large class of quasismooth well-formed del Pezzo weighted hypersurfaces in $\mathbb{P}(a_{1},a_{2},a_{3},a_{4})$. As a corollary we obtain the existence of orbifold K\"ahler--Einstein metrics on many of them, and classify exceptional and weakly exceptional quasismooth well-formed del Pezzo weighted hypersurfaces in $\mathbb{P}(a_{1},a_{2},a_{3},a_{4})$.Comment: 149 pages, one reference adde

Three embeddings of the Klein simple group into the Cremona group of rank three

We study the action of the Klein simple group G consisting of 168 elements on two rational threefolds: the three-dimensional projective space and a smooth Fano threefold X of anticanonical degree 22 and index 1. We show that the Cremona group of rank three has at least three non-conjugate subgroups isomorphic to G. As a by-product, we prove that X admits a Kahler-Einstein metric, and we construct a smooth polarized K3 surface of degree 22 with an action of the group G.Comment: 43 page

The Calabi problem for smooth Fano threefolds

To be published by CUP, LMS Lecture Notes Series 2022Copyright © 2021 The Authors. There are 105 irreducible families of smooth Fano threefolds, which have been classified by Iskovskikh, Mori and Mukai. For each family, we determine whether its general member admits a K¨ahler–Einstein metric or not. We also find all K¨ahler–Einstein smooth Fano threefolds that have infinite automorphism groups.Engineering & Physical Sciences Research Council (EP/056689/1 Calabi conjecture for smooth Fano threefolds); Heilbronn Institute for Mathematical Research (K-stability of smooth Fano 3-folds).https://archive.mpim-bonn.mpg.de/id/eprint/4589/1/mpim-preprint_2021-31.pd