We discuss the central charge in supersymmetric sigma models in two dimensions. The target space is a symmetric Kähler 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.