Suppose $d$ is odd, and consider the super vector space $\mathbb{C}^{d+1|d}$ and its super projectivization $\mathbb{P}^{d|d}$.

In the case $d=1$, a paper of Witten's, concerning the moduli space of super Riemann surfaces, shows that by considering the distribution on the SRS $\mathbb{P}^{1|1}$, as a super skew-symmetric bilinear form on $\mathbb{C}^{2|1}$, $SpO(2|1)$ is the automorphism group of $\mathbb{P}^{1|1}$. In particular, the action of $Sp0(2|1)$ on $\mathbb{P}^{1|1}$ is well-defined.

Is it true that the action of $Sp0(d+1|d)$ is well-defined, i.e is equivalents on all constant multiplies of elements, on $\mathbb{P}^{d|d}$ for any odd $d$?


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