On Mori cone of Bott towers

A Bott tower of height r is a sequence of projective bundlesXr⟶πrXr−1⟶πr−1⋯⟶π2X1=P1⟶π1X0={pt}, where Xi=P(OXi−1⊕Li−1) for a line bundle Li−1 over Xi−1 for all 1≤i≤r and P(−) denotes the projectivization. These are smooth projective toric varieties and we refer to the top object Xr also as a Bott tower. In this article, we study the Mori cone and numerically effective (nef) cone of Bott towers, and we classify Fano, weak Fano and log Fano Bott towers. We prove some vanishing theorems for the cohomology of tangent bundle of Bott towers.