Fano varieties: positivity, K-stability and more
This thesis is about Fano varieties and their properties. We will determine the K-stability of certain singular del Pezzo surfaces and smooth Fano 3-folds, the existence of cylinders in singular del Pezzo surfaces, and also classify higher dimensional Fano varieties with certain properties. In dimension 2, many new examples of K-stable polarized singular del Pezzo surfaces with du Val singular points have been introduced and the existence of polarized cylinders in many of these surfaces has been determined. We also completely solve the K- stability problem for singular del Pezzo surfaces that are index 2 hypersurfaces in weighted projective space. In dimension 3, all deformation families of smooth three-dimensional Fano varieties that contain K-polystable elements have been described. In higher dimensions, a complete classification of smooth Fano varieties of large index that have positive second and third Chern characters has been given, and all rational homogeneous spaces of Picard rank 1 having positive second Chern character have been described. In particular, we prove that the only rational homogeneous spaces of Picard rank 1 with positive second and third Chern characters are projective spaces and quadric hypersurfaces. This thesis also contains few auxiliary results, which are closely related to K- stability of Fano varieties. For instance, for a reduced plane curve of degree d, the sixth worst log canonical threshold that it can have, has been determined.