  The two-dimensional minimal supersymmetric sigma models with homogeneous
target spaces $G/H$ and chiral fermions of the same chirality are revisited. We
demonstrate that the Moore-Nelson consistency condition revealing a global
anomaly in CP(N-1) (with N>2 and ${\mathcal N}=(0,2)$ supersymmetry) due to a
nontrivial first Pontryagin class is in one-to-one correspondence with the
local anomalies of isometries in these models. These latter anomalies are
generated by fermion loop diagrams which we explicitly calculate. In the case
of O}(N) sigma models the first Pontryagin class vanishes, so there is no
global obstruction for the minimal ${\mathcal N}=(0,1)$ supersymmetrization of
these models. We show that at the local level isometries in these models are
anomaly free. Thus, there are no obstructions to quantizing the minimal
${\mathcal N}=(0,1)$ models with the $S^{N-1}= SO(N)/SO(N-1)$ target space.
This also includes CP(1) (equivalent to $S^{2}$) which is an exceptional case
from the CP(N-1) series. We also discuss a relation between the geometric and
gauged formulations of the CP}(N-1) models.
