The Kähler geometry of Bott manifolds

We study the Kähler 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 [3,6], that any stage n Bott manifold M _n admits an extremal Kähler metric. We also give necessary conditions for M _n to admit a constant scalar curvature Kähler 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 Kähler 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 Kähler 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.