of spaces and will be of primary interest in the rest of the paper. The class include any quantum weighted projective line (and in particular all quantum teardrops), since for $n=1$ any weight vector is such that $(\ell_0,\ell_1)=(\ell_1,\ell_0)^{\,\sharp}$.

Quantum weighted projective and lens spaces {#sec:due}
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Fix an integer $n\geq 1$. The coordinate algebra ${\mathcal{O}}(S^{2n+1}_q)$ of the $2n+1$-dimensional quantum sphere is generated by $2(n+1)$ elements $\{z_i,z_i^*\}_{i=0,\ldots,n}$ with relations [@VS91] (see also [@We00]):

\[eq:defrel\] $$\begin{aligned}
z_iz_j &=q^{-1}z_jz_i &&\forall\;0\leq i<j\leq n \;, \\
z_i^*z_j &=qz_jz_i^* &&\forall\;i\neq j \;, \label{eq:defrelB} \\
[z_i^*,z_i] &=(1-q^2)\sum\nolimits_{j=i+1}^n z_jz_j^* &&\forall\;i=0,\ldots,n-1 \;,  \label{eq:defrelC} \\
[z_n^*,z_n] &=0 \;, \qquad \; \\
z_0z_0^*+z_1z_1^* &+\ldots+z_nz_n^*=1 \;. \label{eq:defrelE}\end{aligned}$$

We use the notations of [@DL10]. We have denoted by $q$ the deformation parameter, and assume that $0<q<1$. The original notation of [@VS91] is obtained by setting $q=e^{h/2}$; the generators $x_i$ used in [@HL04] are related to ours by $x_i=z_{n+1-i}^*$ and replacing $q\to q^{-1}$.

Let ${\underline}{\ell}$ be a weight vector and ${\mathrm{U}}(1) = \{ t\in{\mathbb{C}}, \, |t|=1\}$. An action by $*$-automorphisms $\alpha:{\mathrm{U}}(1)\to{\mathrm}{Aut}\,{\mathcal{O}}(S^{2n+1}_q)$ is defined on generators by: $$\label{eq:act}
\alpha_t(z_i):=t^{\ell_i}z_i \;,\qquad\forall\;i=0,\ldots,n \, .$$ Invariant elements form a $*$-subalgebra $$\label{eq:WP}
{\mathcal{O}}({\mathbb{P}}_q({\underline}{\ell})):=\big\{a\in{\mathcal{O}}(S^{2n+1}_q):\alpha_t(a)=a \;\forall\; t\in {\mathrm{U}}(1)\big\} \;.$$ The virtual underlying quantum space is called *quantum weighted (complex) projective space* 