Jongbaek Song

Iterated weighted projective space fibrations and toric orbifolds

Introduction

Iterated weighted projective space fibrations and toric orbifolds. Generalize Bott towers to orbifolds using weighted projective towers. Explore toric topology & derive explicit criteria for toric orbifolds over products of simplices to admit this structure.

Abstract

We generalize classical generalized Bott towers to orbifolds using weighted projectivizations of line bundles, which we call \emph{weighted projective towers}. From the perspective of toric topology, such a space can be constructed from a product of simplices with a rational characteristic function on it. However, such a construction gives an orbifold fibration in general. Our main theorem provides explicit criteria for when a toric orbifold over a product of simplices admits a structure of a weighted projective tower.

Review

This paper introduces the concept of "weighted projective towers," a significant generalization of classical generalized Bott towers adapted for orbifolds through the iterated weighted projectivization of line bundles. The authors also establish a crucial connection to toric topology, framing these spaces as constructs arising from products of simplices endowed with rational characteristic functions, which naturally lead to orbifold fibrations. The main achievement is the provision of explicit criteria that determine precisely when a toric orbifold, derived from such a simplicial product, can be structured as a weighted projective tower. This work effectively bridges advanced concepts in algebraic geometry and toric topology, offering a novel framework for the study of complex geometric objects. The generalization of Bott towers to the orbifold setting is a substantial theoretical advancement. Classical Bott towers are foundational to understanding flag varieties and their topological invariants, and extending this theory to orbifolds opens up new avenues for exploring more general classes of spaces. The notion of "weighted projective towers" further enriches this generalization by incorporating weighted projectivization, suggesting a broader and potentially more intricate array of structures. The interdisciplinary approach, leveraging insights from both algebraic geometry and the combinatorial methods of toric topology, is a particular strength. The promise of "explicit criteria" is especially noteworthy, indicating that the paper delivers concrete and applicable conditions, making the results highly practical for researchers attempting to identify or construct these specific orbifold structures. This research has considerable potential to influence our understanding in both orbifold geometry and toric topology. The explicit conditions for constructing weighted projective towers on toric orbifolds could serve as a powerful tool for classification, construction, and the analysis of invariants for these complex spaces. Future work might explore the homological or K-theoretic properties of these towers, or investigate their role in other areas of geometry, such as mirror symmetry for orbifolds. Overall, the paper appears to lay robust groundwork for continued exploration into the geometry and topology of these generalized Bott towers, promising to stimulate further research in related fields.

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