A 3,4-isoprene 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: (A) wherein R<1> and R<2> independently represent an alkyl group, a cyclohexyl group, an aryl group or an aralkyl group; R<3> represents an alkyl group, an alkenyl group, an alkynyl group, an aryl group, an aralkyl group, an aliphatic, aromatic or cyclic amino, phosphino, boryl, alkylthio or arylsulfur, or an alkoxy or aryloxy group; M represents a rare earth element selected from La to Lu, provided that the rare earth element is not Sc, Y or Pm; Q<1> and Q<2> independently represent a monoanionic ligand; L represents a neutral Lewis base; and w represents an integer of 0 to 3.