3,4-isoprene-based polymer having high isotacticity can be produced by polymerizing an isoprene compound using a complex represented by the general formula (A) and a catalyst activator: wherein R1 and R2 independently represent an alkyl group, a cyclohexyl group, an aryl group or an aralkyl group; R3 represents an alkyl group, an alkenyl group, an alkynyl group, an aryl group, an aralkyl group, an aliphatic, aromatic or cyclic amino group, a phosphino group, a boryl group, an alkylthio or arylthio group, or an alkoxy or aryloxy group; M represents a rare earth element selected from Sc, Y, and La to Lu with promethium (Pm) excluded; Q1 and Q2 independently represent a monoanionic ligand; L represents a neutral Lewis base; and w represents an integer of 0 to 3.