Bott Towers, Crosspolytopes and Torus Actions

Abstract

Abstract. We study the geometry and topology of Bott towers in the context of toric geometry. We show that any kth stage of a Bott tower is a smooth projective toric variety associated to a fan arising from a crosspolytope; conversely, we prove that any toric variety associated to a fan obtained from a crosspolytope actually gives rise to a Bott tower. The former leads us to a description of the tangent bundle of the kth stage of the tower, considered as a complex manifold, which splits into a sum of complex line bundles. Applying Danilov-Jurkiewicz theorem, we compute the cohomology ring of any kth stage, and by way of construction, we provide all the monomial identities defining the related affine toric varieties.

On Mori cone of Bott towers

“…In this section we recall toric varieties (see [CLS11]) and Bott towers (see [Civ05] and [VT15]). We work throughout the article over the field C of complex numbers.…”

“…For each 1 ≤ i ≤ r, X i is a smooth projective toric variety (see [Civ05,Theorem 22]). Consider the points [1 : 0] and [0 : 1] in P 1 , we call them the south pole and the north pole respectively.…”

“…To describe these results we need some notation. It is known that a Bott tower {(X i , π i ) : 1 ≤ i ≤ r} is uniquely determined by an upper triangular matrix M r with integer entries, defined via the first Chern class of the line bundle L i−1 on X i−1 , where X i = P(O X i−1 ⊕ L i−1 ) for 1 ≤ i ≤ r (see [GK94,Section 2.3], [Civ05] and [VT15,Section 7.8] where β ij 's are integers. Define for 1 ≤ i ≤ r, η + i := {r ≥ j > i : β ij > 0} and η − i := {r ≥ j > i : β ij < 0}.…”

The Kähler geometry of Bott manifolds

“…, n}, labelled by the Bott tower structure, and only opposite pairs have empty intersection. The divisors thus have the structure of an n-cross or cross-polytope (the dual of an n-cube) and the cones of the fan are therefore cones over the faces of an n-cross in t with vertex set S, the rays of the fan [Civ05].…”

A note on toric degeneration of a Bott–Samelson–Demazure–Hansen variety

“…Recall that the Bott towers bijectively correspond to the upper triangular matrices with integer entries (see [Civ05,Section 3]). Here the upper triangular matrix Mw corresponding to Yw is given by…”