On Mori cone of Bott towers

Abstract: A Bott tower of height $r$ is a sequence of projective bundles $$X_r \overset{{\pi_r}}\longrightarrow X_{r-1} \overset{\pi_{r-1}}\longrightarrow \cdots \overset{\pi_2}\longrightarrow X_1=\mathbb P^1 \overset{\pi_1} \longrightarrow X_0={pt}, $$ where $X_i=\mathbb P (\mathcal O_{X_{i-1}}\oplus \mathcal L_{i-1})$ for a line bundle $\mathcal L_{i-1}$ over $X_{i-1}$ for all $1\leq i\leq r$ and $\mathbb P(-)$ denotes the projectivization. These are smooth projective toric varieties and we refer to the top object $X_{r}$ 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.