  We investigate the singlet $g_T^{(0)}$ and isovector $g_T^{(3)}$ tensor
charges of the nucleon, which are deeply related to the first moment of the
leading twist transversity quark distribution $h_1(x)$, in the chiral
quark-soliton model. With rotational $O(1/N_c) $ corrections taken into
account, we obtain $g_T^{(0)}=0.69$ and $g_T^{(3)}=1.45$ at a low normalization
poin of several hundreds MeV.
  Within the same approximation and parameters the model yields
$g_A^{(0)}=0.36$, $g_A^{(3)}=1.21$ for axial charges and correct
octet--decuplet mass splitting. We show how the chiral quark-soliton model
interpolates between the nonrelativistic quark model and the Skyrme model.
