Weighted Projective Spaces

• Weighted Projective Spaces are complex algebraic and topological spaces defined as quotients by weighted scalar actions.
• They exhibit intricate orbifold singularities and utilize normalized weight vectors and p-content for precise classification.
• Homotopy equivalence and Mislin genus rigidity provide deeper insights into their structural, geometric, and topological properties.

A weighted projective space is a fundamental class of complex algebraic and topological spaces defined as a quotient of a vector space by a weighted scalar action. These spaces play key roles across algebraic geometry, toric geometry, cohomological theory, and singularity classification. Weighted projective spaces generalize classical projective space by allowing unequal weightings on homogeneous coordinates. This modification induces rich structures, including orbifold singularities and refined equivalence classifications, governed by the combinatorics of the weights and their prime decompositions.

1. Topological and Algebraic Definitions

A weighted projective space of dimension $n$ is specified by a weight vector $x = (x_0, x_1, ..., x_n)$, with each $x_i > 0$. Consider the standard sphere $S^{2n+1} \subset \mathbb{C}^{n+1}$ or the punctured affine space $\mathbb{C}^{n+1} \setminus \{0\}$. The $S^1$ (or $\mathbb{C}^\times$ as appropriate) acts by

$$g \cdot (z_0, ..., z_n) = (g^{x_0}z_0, ..., g^{x_n}z_n).$$

The weighted projective space is then the quotient: $$P(x) = S^{2n+1}/S^1(x) \cong (\mathbb{C}^{n+1}\setminus \{0\}) / \mathbb{C}^\times(x).$$ These spaces are complex projective varieties, and their structure is inherently determined by the weight vector. The weights can always be "normalized"—for each prime $p$, one can replace $x$ by a coordinate-wise reduced vector ensuring that at least two entries are not divisible by $p$, with the resulting $P(x)$ remaining isomorphic as a variety and homeomorphic as a space.

2. Classification: Homeomorphism and Algebraic Isomorphism

Classification up to homeomorphism (and equivalently, algebraic variety isomorphism) rests completely on the normalized weight vector:

Theorem 1.1

P$(x)$ and P$(x')$ are homeomorphic (topologically) and isomorphic (in the sense of algebraic varieties) if and only if their normalized weight vectors coincide up to permutation. That is,

$$\exists \sigma \in S_{n+1}: (x_0, ..., x_n)_{norm} = (x'_{\sigma(0)}, ..., x'_{\sigma(n)})_{norm}.$$

This result recovers and topologically justifies Al Amrani's algebraic classification, which is originally formulated using K-theoretical invariants. In particular, the fine structure of the weight vector, after normalization, is a complete invariant for these equivalence relations.

3. Homotopy Classification and p-Content Invariants

Homotopy equivalence of weighted projective spaces is governed by the more subtle “$p$-content” invariants:

Given a normalized weight vector $x$ and each prime $p$, the $p$-content $p(x)$ is the vector whose $i$-th entry is the maximal power of $p$ dividing $x_i$. The main result here states:

Theorem 1.2

Two weighted projective spaces $P(x)$, $P(x')$ are homotopy equivalent if and only if, for every prime $p$, the $p$-contents of their normalized weight vectors agree up to reordering.

This result reveals that homotopy theory “forgets” certain arithmetic details: there exist weighted projective spaces that are not homeomorphic (nor isomorphic as varieties) but are homotopy equivalent because their weight data agree at the level of $p$-contents for all primes. The underlying argument is founded on Kawasaki’s explicit computation of the integral cohomology ring $H^*(P(x);\mathbb{Z})$, which is sensitive only to the $p$-content.

4. Mislin Genus Rigidity and Localization Methods

The Mislin genus $\mathcal{G}(X)$ of a finite CW complex $X$ is the set of finite-type simply connected complexes $Y$ whose localizations at every prime $p$ (including $0$) are homotopy equivalent to $X_{(p)}$. The paper demonstrates:

Theorem 1.3

The Mislin genus of any weighted projective space is rigid, i.e., every $Y$ with $Y_{(p)} \simeq P(x)_{(p)}$ for every $p$ is homotopy equivalent to $P(x)$ itself.

Rigidity is proven by explicit analysis of self-maps: $$H^2(h; \mathbb{Z}_{(p)})(\eta_1) = \deg(h) \cdot \eta_1,$$ with the criterion that $h$ is a homotopy equivalence if and only if $\deg(h)$ is a unit in $\mathbb{Z}_{(p)}$. The proof leverages localization and pullback properties to rule out exotic genus representatives.

5. Key Formulae and Computations

Specific structural and computational results central to the classification are as follows:

Structure/Formula	Expression
Weighted $S^1$ action	$$g\cdot(z_0,...,z_n) = (g^{x_0}z_0, ..., g^{x_n}z_n)$$
Presentation	$P(x) = S^{2n+1}/S^1(x)$ or $$(\mathbb{C}^{n+1}\setminus\{0\})/\mathbb{C}^\times(x)$$
$p$-content	$p(x) = (p^{k_0}, ..., p^{k_n})$, $k_i = \max\{k : p^k| x_i\}$
Degree of self-map $h$	$$H^2(h;\mathbb{Z}_{(p)})(\eta_1) = \deg(h)\cdot \eta_1$$
Homotopy classification	$P(x) \simeq P(x') \iff p(x) = p(x')$ for all $p$, up to order

These reflect both the quotient topological structure and the cohomological data that encode the weight vector’s arithmetic.

6. Geometric and Topological Implications

The normalization and $p$-content invariants not only direct classification but also inform geometric structure, singular stratification, and cohomology:

• Spaces with the same $p$-content have isomorphic generalized cohomology rings for all complex-oriented theories.
• While homeomorphism (and isomorphism as varieties) detect fine weight data, homotopy equivalence “identifies” spaces with the same “prime filtration” of their weight vector.
• The rigidity of the Mislin genus situates weighted projective spaces among those finite spaces completely determined by their localizations, paralleling classical projective spaces but with richer arithmetic features.

7. Broader Context and Significance

These classification results establish a comprehensive connection between the combinatorics of the weight vector and both algebraic and topological invariants. They clarify the geometric meaning of existing algebraic classifications (e.g., those of Al Amrani) and reveal the limits of homotopy theory in detecting arithmetic data. The rigidity of the Mislin genus constrains possible exotic behavior in the homotopy category and ensures that local-to-global principles are particularly tight for this class of spaces.

The results position weighted projective spaces as paradigmatic objects at the intersection of algebraic geometry, equivariant topology, and prime-localized homotopy theory, with implications for computations in toric geometry, equivariant cohomology, and moduli theory.

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References:

• "The classification of weighted projective spaces" (Bahri et al., 2011)