Abstract

The main objects of study are smooth toric Fano varieties. A classical result of Batyrev implies that given such a variety $X$,  a particular relation among a subset of primitive generators of the fan of $X$ exists, called the centrally symmetric primitive relation. Such relations in turn describe the geometry of $X$. We define a new invariant called the minimal projective bundle dimension $m(X)$ depending on the length of such a relation which has wide applications.  In this talk, I will detail the classification of toric Fano manifolds done using this invariant. In addition to this, I would like to explore the connections between the invariant $m(X)$ and K-stability of $X$.   The work on the invariant, done to explore the nature of higher Fano condition is joint work with Carolina Araujo, Roya Beheshti, Ana-Maria Castravet, Kelly Jabbusch, Svetlana Makarova and Enrica Mazzon.