  In the context of the star formation through the fragmentation of an
extremely metal-deficient protogalactic cloud, the gravitational collapse of
filamentary gas clouds is explored with H$_2$ and HD chemistry. It is found by
1D hydrodynamical simulations that the cloud evolution is prescribed mainly by
the initial density ($n_0$) and H$_2$ abundance ($x_{\rm H_2,0}$). In
particular, it turns out that the evolution of low-density filaments ($n_0
\lesssim 10^5$ cm$^{-3}$) bifurcates at a critical H$_2$ abundance of $x_{\rm
H_2,cr}\simeq 3\times 10^{-3}$, beyond which HD cooling overwhelms H$_2$
cooling. The numerical results indicate that the stellar IMF is likely to be
double-peaked and deficient in sub-solar mass stars, where the high mass peak
of the IMF is around $10M_\odot$ or $10^2M_\odot$, dependently on the initial
density and H$_2$ abundance. If the gas in protogalactic clouds is photoionized
by UV radiation or shock-heated, the H$_2$ abundance could exceed
  $x_{\rm H_2,cr}\simeq 3\times 10^{-3}$ by H$^-$ reactions. Then, the high
mass peak would be $O(10) M_\odot$.
