# On Mori cone of Bott towers

## TL;DR

This paper investigates the Mori cone and nef cone of Bott towers, classifies their Fano properties, and proves vanishing theorems for tangent bundle cohomology, advancing understanding of their geometric and algebraic structure.

## Contribution

It provides a detailed analysis of the Mori and nef cones of Bott towers, classifies Fano, weak Fano, and log Fano Bott towers, and establishes new vanishing theorems for tangent bundle cohomology.

## Key findings

- Classification of Fano, weak Fano, and log Fano Bott towers.
- Explicit description of Mori and nef cones for Bott towers.
- Vanishing theorems for tangent bundle cohomology.

## 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.