On the geometry of some Fano and hyperkähler varieties via homogeneous spaces

In this thesis, we employ tools from the theory of homogeneous varieties to investigate the geometry of certain Fano and hyperkähler varieties. The work is structured into four main chapters. The first chapter provides a concise overview of the necessary background material and revisits classical techniques used in the study of homogeneous varieties. It introduces the main methods for computing Hodge numbers and discusses the rationale for choosing homogeneous varieties (particularly Grassmannians) as our primary framework. In the second chapter, we examine 170 families of quiver flag zero-locus Fano fourfolds, as described by Kalashnikov. We interpret these manifolds as zero loci of sections of homogeneous vector bundles over homogeneous varieties, and we present both birational and biregular descriptions of all 170 families. The third chapter proves that all Schur functors of the restriction of the quotient bundle on Gr(6, 10) to a very general Debarre–Voisin fourfold are modular and polystable vector bundles. We further show that these bundles are atomic if and only if they correspond to symmetric powers of the restriction of the quotient bundle. In addition, we compute the Ext-groups of various modular vector bundles on a very general Debarre–Voisin fourfold, exhibiting examples with first Ext-groups of dimensions 20 and 40. Finally, in the fourth chapter, we apply the techniques and results developed in the first chapter to explore the geometry of a remarkable Fano fourfold of K3 type, namely the Küchle c5 fourfold.