Some K\"ahler structures on products of 2-spheres

TL;DR

This paper studies a family of K"ahler structures on products of 2-spheres derived from complex Bott manifolds, classifying them via Bott diagrams and analyzing their Chern classes.

Contribution

It introduces Bott diagrams as a classification tool for these K"ahler structures and shows they share identical Chern classes.

Findings

All structures have the same Chern classes.

Bott diagrams classify the structures up to biholomorphism.

Generalization of Hirzebruch surfaces to higher dimensions.

Abstract

We consider a family of K\"ahler structures on products of 2-spheres, arising from complex Bott manifolds. These are obtained via iterated $\mathbb P^1$-bundle constructions, generalizing the classical Hirzebruch surfaces. We show that the resulting K\"ahler structures all have identical Chern classes. We construct Bott diagrams, which are rooted forests with an edge labelling by positive integers, and show that these classify these K\"ahler structures up to biholomorphism.