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The classification of weighted projective spaces. (English) Zbl 1271.55006

Weighted projective spaces are quotients of \(S^{2n+1} \subset \mathbb{C}^{n+1}\) by a linear action of \(S^1\), or equivalently of \(\mathbb{C}^{n+1}\smallsetminus\{0\}\) by a linear action \(\mathbb{C}^\times\). Consequently, a weighted projective space simultaneously carries the structure of a smooth orbifold and a projective toric variety. The \(S^1\)- or \(\mathbb{C}^\times\)-action is indicated by a weight vector in \(\mathbb{Z}^{n+1}\). The authors give classifications of weighted projective spaces up to homeomorphism and homotopy equivalence in terms of properties of the weight vector.
The first main theorem demonstrates that two weighted projective spaces are homeomorphic if and only if they are isomorphic as algebraic varieties. Using the classification of weighted projective spaces up to isomorphism of algebraic varieties by A. Al Amrani [K-Theory 2, No. 5, 559–578 (1989; Zbl 0697.14004)], the authors give a simple criterion on the weight vectors that indicates whether two weighted projective spaces are homeomorphic. Specifically, scaling all \(n+1\) or any \(n\) weights (without leaving \(\mathbb{Z}^{n+1}\)) does not change the homeomorphism class of the weighted projective space, so one may assume without loss of generality that for any prime \(p\), at least two weights are not divisible by \(p\). Under this assumption, two weighted projective spaces are homeomorphic if and only if their weight vectors coincide up to a permutation of the weights.
The second result demonstrates a simple criterion on the weight vector to determine whether two weighted projective spaces are homotopy equivalent. For a prime \(p\), the \(p\)-content of a weight vector is the vector made up of the highest powers of \(p\) that divide each weight. Under the divisibility assumption above, two weighted projective spaces are homotopy equivalent if and only if their weight vectors have the same \(p\)-content up to a permutation of the weights. This is proven by demonstrating that the Mislin genus of a weighted projective space is rigid, i.e. contains only the class of the entire space.

MSC:

55P15 Classification of homotopy type
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
55P60 Localization and completion in homotopy theory
57R18 Topology and geometry of orbifolds

Citations:

Zbl 0697.14004
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