  We discuss the central charge in supersymmetric ${\cal N}=2$ sigma models in
two dimensions. The target space is a symmetric K\"ahler manifold, CP$(N-1)$ is
an example. The U(1) isometries allow one to introduce twisted masses in the
model. At the classical level the central charge contains Noether charges of
the U(1) isometries and a topological charge which is an integral of a total
derivative of the Killing potentials. At the quantum level the topological part
of the central charge acquires anomalous terms. A bifermion term was found
previously, using supersymmetry which relates it to the superconformal anomaly.
We present a direct calculation of this term using a number of regularizations.
We derive, for the first time, the bosonic part in the central charge anomaly.
We construct the supermultiplet of all anomalies and present its superfield
description. We also discuss a related issue of BPS solitons in the CP(1) model
and present an explicit form for the curve of marginal stability.
