The K\"ahler geometry of Bott manifolds

TL;DR

This paper investigates the K"ahler geometry of Bott manifolds, proving the existence of extremal metrics for all stages and exploring conditions for constant scalar curvature, with detailed results for stage 3 and connections to other geometric structures.

Contribution

It demonstrates that all stage n Bott manifolds admit extremal K"ahler metrics and provides necessary conditions for constant scalar curvature, expanding understanding of their geometric properties.

Findings

All stage n Bott manifolds admit extremal K"ahler metrics.

Necessary conditions for constant scalar curvature K"ahler metrics are identified.

Explicit examples and relations with c-projective geometry are provided for stage 3 Bott manifolds.

Abstract

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.