Maximal log Fano pairs as generalised Bott towers

Log Fano varieties are natural generalizations of Fano varieties. They are defined as pairs $(X, D)$ such that $-K_X-D$ is ample and $D$ is a divisor called a boundary. We consider the case of smooth projective $X$ and reduced divisor D with simple normal crossings. Such pairs were studied by H. Maeda, Takao and Kento Fujita, and others. If in the above definition we put $D = 0$ then we recover the classical definition of a Fano variety. We will study the opposite case of 'large enough' boundary divisor D. More precisely, we will show that if D has maximal possible number of components (such log Fano pairs we call maximal) then the geometry of X, including the Mori cone and extremal contractions, can be explicitly described. It turns out that such pairs $(X, D)$ are toric and moreover, $X$ admits the structure of a generalized Bott tower. This means that X is an iterated projective bundle over a point. If time permits, we will discuss how maximal log Fano pairs are related to semistable degenerations of Fano varieties. The talk is based on a joint work with J. Moraga.