Filtered cohomological rigidity of Bott towers (Q436059)

A Bott tower is a sequence of \({\mathbb{C}} P^1\)-bundles NEWLINE\[NEWLINE B_n\to B_{n-1}\to\cdots\to B_1\to B_0, NEWLINE\]NEWLINE where \(B_0\) is a point, and each \({\mathbb{C}} P^1\)-bundle \(\pi_i:B_i\to B_{i-1}\) is the projectivization of the sum of two complex line bundles over \(B_{i-1}\). The so-called Bott manifold \(B_n\) is a smooth projective toric variety.NEWLINENEWLINEVia \(\pi_i^*\), the cohomology \(H^*(B_{i-1})\) (with integer coefficients) embeds into \(H^*(B_i)\), and thus a Bott tower as above naturally defines a filtration of \(H^*(B_n)\). Motivated by the cohomological rigidity problem for toric manifolds, the author proves that any filtered isomorphism between the cohomologies of two Bott manifolds is induced by an isomorphism of the Bott towers.NEWLINENEWLINEThe proof goes by induction on the height \(n\) of the Bott towers; the filtered isomorphisms are understood via the fact that the \(H^*(B_{i-1})\)-algebra structure of \(H^*(B_i)\) is determined by the Chern classes of the rank two bundle that defines \(\pi_i\).