The Kähler geometry of Bott manifolds

Abstract

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 [ACGT04, ACGT11], 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.

Sasaki–Einstein metrics on a class of 7-manifolds

“…Following [GK94] and [BCTF18] we consider Bott towers which in arbitrary dimension is represented by a lower triangular unipotent matrix A over Z. Here we deal only with stage 3 Bott towers, so the matrix A in [GK94,BCTF18] takes the form with a, b, c ∈ Z. The Bott manifold can be realized as the quotient of S 3 × S 3 × S 3 by the T 3 action…”

“…M 3 (a, b, c) can be viewed as the total space of CP 1 bundle over the Hirzebruch surface H a , and also a bundle of Hirzebruch surfaces over CP 1 with fiber H c . Bott towers form the object set BT 0 of a groupoid whose morphisms BT 1 are biholomorphisms [BCTF18], and elements of the quotient space BT 0 /T 1 are identfied with biholomorphism classes of Bott manifolds. Since Bott manifolds are toric, they are described by a fan, and it follows from the Bott tower description (3) that the fan of the Bott tower M 3 (a, b, c) is described by the primitive collections (cf.…”

“…However, not all elements of BC 3 are induced by equivalences. We refer to [BCTF18] and references therein for details.…”

“…Recall (Definition 1.6 of [BCTF18]) that the holomorphic twist of an n dimensional Bott tower is the number t ∈ {0, . .…”

“…We give an explicit construction of toric Sasaki-Einstein (SE) 7-manifolds which can be represented as S 1 orbibundles over 2-twist stage 3 Bott orbifolds. All of these are obtained by adding orbifold structures to certain stage 3 Bott manifolds which were studied in [BCTF18]. The 7-manifolds all have the rational cohomology of the 2-fold connected sum 2(S 2 ×S 5 ) and are generalizations of the SE 7-manifolds given by Theorem 1.2 of [BTF15].…”