Topics on the geometry of homogeneous spaces

Abstract

This is a survey paper about a selection of results in complex algebraic geometry that appeared in the recent and less recent litterature, and in which rational homogeneous spaces play a prominent rôle. This selection is largely arbitrary and mainly reflects the interests of the author.Rational homogeneous varieties are very special projective varieties, which appear in a variety of circumstances as exhibiting extremal behavior. In the quite recent years, a series of very interesting examples of pairs (sometimes called Fourier-Mukai partners) of derived equivalent, but not isomorphic, and even non birationally equivalent manifolds have been discovered by several authors, starting from the special geometry of certain homogeneous spaces. We will not discuss derived categories and will not describe these derived equivalences: this would require more sophisticated tools and much ampler discussions. Neither will we say much about Homological Projective Duality, which can be considered as the unifying thread of all these apparently disparate examples. Our much more modest goal will be to describe their geometry, starting from the ambient homogeneous spaces.In order to do so, we will have to explain how one can approach homogeneous spaces just playing with Dynkin diagram, without knowing much about Lie theory. In particular we will explain how to describe the VMRT (variety of minimal rational tangents) of a generalized Grassmannian. This will show us how to compute the index of these varieties, remind us of their importance in the classification problem of Fano manifolds, in rigidity questions, and also, will explain their close relationships with prehomogeneous vector spaces.We will then consider vector bundles on homogeneous spaces, and use them to construct interesting birational transformations, including important types of flops: the Atiyah and Mukai flops, their stratified versions, also the Abuaf-Segal and Abuaf-Ueda flops; all these beautiful transformations are easily described in terms of homogenenous spaces. And introducing sections of the bundles involved, we will quickly arrive at several nice examples of Fourier-Mukai partners.We will also explain how the problem of finding crepant resolutions of orbit closures in prehomogeneous spaces is related to the construction of certain manifolds with trivial canonical class. This gives a unified perspective over classical constructions by Reid, Beauville-Donagi and Debarre-Voisin of abelian and hyperKähler varieties, naturally embedded into homogeneous spaces. The paper will close on a recent construction, made in a similar spirit, of a generalized Kummer fourfold from an alternating three-form in nine variables.