  We summarize our studies on neutrino-driven nucleosynthesis in He shells of
early core-collapse supernovae with metallicities of $Z\lesssim
10^{-3}\,Z_\odot$. We find that for progenitors of $\sim 11$--$15\,{\rm
M}_\odot$, the neutrons released by $^4{\rm He}(\bar{\nu}_e,e^+n)^3{\rm H}$ in
He shells can be captured to produce nuclei with mass numbers up to $A \sim
200$. This mechanism is sensitive to neutrino emission spectra and flavor
oscillations. In addition, we find two new primary mechanisms for
neutrino-induced production of $^{9}$Be in He shells. The first mechanism
produces $^9$Be via $^7{\rm Li}(n,\gamma)^8{\rm Li}(n,\gamma)^9{\rm
Li}(e^-\bar{\nu}_e)^9{\rm Be}$ and relies on a low explosion energy for its
survival. The second mechanism operates in progenitors of $\sim 8\,{\rm
M}_\odot$, where $^9$Be can be produced directly via $^7{\rm Li}(^3{\rm
H},n_0)^9{\rm Be}$ during the rapid expansion of the shocked He-shell material.
The light nuclei $^7$Li and $^3$H involved in these mechanisms are produced by
neutrino interactions with $^4$He. We discuss the implications of
neutrino-induced nucleosynthesis in He shells for interpreting the elemental
abundances in metal-poor stars.
