  We compute the quark and antiquark transversity distributions in the nucleon
at a low normalization point of 600 MeV in the large-$N_c$ limit, where the
nucleon can be described as a soliton of an effective chiral theory (chiral
quark-soliton model). The flavor-nonsinglet distributions, $\delta u(x) -
\delta d(x)$ and $\delta\bar u(x) - \delta\bar d(x)$, appear in leading order
of the $1/N_c$-expansion, while the flavor-singlet distributions, $\delta u(x)
+ \delta d(x)$ and $\delta\bar u(x) + \delta\bar d(x)$, are non-zero only in
next-to-leading order. The transversity quark and antiquark distributions are
found to be significantly different from the longitudinally polarized
distributions $\Delta u (x) \pm \Delta d (x)$ and $\Delta\bar u (x) \pm
\Delta\bar d (x)$, respectively, in contrast to the prediction of the naive
non-relativistic quark model. We show that this affects the predictions for the
spin asymmetries in Drell-Yan pair production in transversely polarized pp and
ppbar collisions.
