Title:

Deformations of QFano 3folds and weak Fano manifolds

Fano varieties are one of important classes in the classi cation of algebraic varieties. In this thesis, we mainly study problems on deformations of Fano varieties motivated by the classi cation problems. In particular, we study Fano 3folds with terminal singularities and weak Fano manifolds. In Chapter 2, we prepare necessary notions on deformation theory and singularities. We also explain about the orbifold RiemannRoch formula and computation of numerical data of a K3 surface with Du Val singularities and a QFano 3fold. In Chapter 3, we study the deformation theory of a QFano 3fold with only terminal singularities. First, we show that the Kuranishi space of a QFano 3fold is smooth. Second, we show that every QFano 3fold with only "ordinary" terminal singularities is Qsmoothable, that is, it can be deformed to a QFano 3fold with only quotient singularities. Finally, we prove Qsmoothability of a QFano 3fold assuming the existence of a Du Val anticanonical element. As an application, we get the genus bound for primary QFano 3folds with Du Val anticanonical elements. In Chapter 4, we prove that a weak Fano manifold has unobstructed deformations. For a general variety, we investigate conditions under which a variety is necessarily obstructed. In Chapter 5, we investigate a certain coboundary map associated to a 3fold terminal singularity which is important in the study of deformations of singular 3folds. We determine when this map vanishes. As an application, we prove that almost all QFano 3folds have Qsmoothing. We also treat the Qsmoothability problem on QCalabiYau 3folds. In Chapter 6, we study deformations of a pair of a QFano 3fold X with its elephant D E KXZ. We prove that, if X has only quotient singularities and there exists D with only isolated singularities, there is a deformation X > [triangle]1 of X over a unit disc such that KXt has a Du Val element for t E [triangle]1\0. We also give several examples of QFano 3folds without Du Val elephants.
