Fano varieties: positivity, K-stability and more
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Date
11/04/2022Author
Viswanathan, Nivedita
Metadata
Abstract
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.