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We mainly show the nondegeneracy of positive bubble solutions for generalizedenergy-critical Hartree equations (NLH) \begin{equation*}  -{\Delta u}\sts{x}  -{\bm\alpha}\sts{N,\lambda}  \int_{\R^N}  { \frac{ u^{p}\sts{y}}{\pabs{\,x-y\,}{\lambda}} }\diff{y}\,  u^{p-1}\sts{x}  =0,\quad x\in \R^N \end{equation*} where $u(x)$ is a real-valued function,$N\geq 3$, $0<\lambda<N$, $p=\frac{2N-\lambda}{N-2}$ and${\bm\alpha}\sts{N,\lambda}$ is a normalized constant. It generalizes thenondegeneracy results of positive bubble solutions in \cite{DY2019dcds,GWY2020na, LTX2021} to the full range $0<\lambda<N$, and confirms an opennondegeneracy problem in \cite{GMYZ2022cvpde}. The key observation is that theweighted pushforward map $\mathcal{S}_*$ related to the stereographicprojection $\mathcal{S}$ is one-to-one map between the null space of thelinearized operator and the spherical harmonic function subspace$\mathcal{H}_1^{N+1}$ of order one.

Nondegeneracy of positive bubble solutions for generalized energy-critical Hartree equations