Abstract:
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.
