  We carry out a modulation instability (MI) analysis in nonlinear complex
parity-time (PT) symmetric periodic structures. All the three regimes defined
by the PT-symmetry breaking point or threshold, namely, below threshold, at
threshold and above threshold are discussed. It is found that MI exists even
beyond the PT-symmetry threshold indicating the possible existence of solitons
or solitary waves, in conformity with some recent reports. We find that MI does
not exist at the PT-symmetry breaking point in the case of normal dispersion
below a certain nonlinear threshold. However, in the case of anomalous
dispersion regime, MI does exist even at the PT-symmetry breaking point.
