Deformations of Q-Fano 3-folds and weak Fano manifolds
arxiv: Mathematics::Algebraic Geometry | Quantitative Biology::Biomolecules | Mathematics::Symplectic Geometry
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 3-folds with terminal singularities and weak Fano manifolds.\ud \ud In Chapter 2, we prepare necessary notions on deformation theory and singularities. We also explain about the orbifold Riemann-Roch formula and computation of numerical data of a K3 surface with Du Val singularities and a Q-Fano 3-fold.\ud \ud In Chapter 3, we study the deformation theory of a Q-Fano 3-fold with only terminal singularities. First, we show that the Kuranishi space of a Q-Fano 3-fold is smooth. Second, we show that every Q-Fano 3-fold with only "ordinary" terminal singularities is Q-smoothable, that is, it can be deformed to a Q-Fano 3-fold with only quotient singularities. Finally, we prove Q-smoothability of a Q-Fano 3-fold assuming the existence of a Du Val anticanonical element. As an application, we get the genus bound for primary Q-Fano 3-folds with Du Val anticanonical elements.\ud \ud 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.\ud \ud In Chapter 5, we investigate a certain coboundary map associated to a 3-fold terminal singularity which is important in the study of deformations of singular 3-folds. We determine when this map vanishes. As an application, we prove that almost all Q-Fano 3-folds have Q-smoothing. We also treat the Q-smoothability problem on Q-Calabi-Yau 3-folds.\ud \ud In Chapter 6, we study deformations of a pair of a Q-Fano 3-fold 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 Q-Fano 3-folds without Du Val elephants.