The K\"ahler geometry of Bott manifolds

We study the K"ahler geometry of stage n Bott manifolds, which can be viewed as $n$-dimensional generalizations of Hirzebruch surfaces. We show, using a simple induction argument and the generalized Calabi construction from [ACGT04,ACGT11], that any stage n Bott manifold $M_n$ admits an extremal K"ahler metric. We also give necessary conditions for $M_n$ to admit a constant scalar curvature K"ahler metric. We obtain more precise results for stage 3 Bott manifolds, including in particular some interesting relations with c-projective geometry and some explicit examples of almost K"ahler structures. To place these results in context, we review and develop the topology, complex geometry and symplectic geometry of Bott manifolds. In particular, we study the K"ahler cone, the automorphism group and the Fano condition. We also relate the number of conjugacy classes of maximal tori in the symplectomorphism group to the number of biholomorphism classes compatible with the symplectic structure.