Abstract: Catalyst systems for the polymerization of C.sub.2 -C.sub.10 -alk-1-enes contain, as active components,a) a metallocene complex of the general formula I ##STR1## where M is a metal of the subgroup III, IV or v of the Periodic Table of elements or a metal selected from the group consisting of the lanthanides,X is fluorine, chlorine, bromine, iodine, hydrogen, C.sub.1 -C.sub.10 -alkyl, C.sub.6 -C.sub.15 -aryl or --OR.sup.7,R.sup.7 is C.sub.1 -C.sub.10 -alkyl, C.sub.6 -C.sub.15 -aryl, alkylaryl, arylalkyl, fluoroalkyl or fluoroaryl, in each case having 1 to 10 carbon atoms in the alkyl radical and 6 to 20 carbon atoms in the aryl radical,n is the valency of M minus two,R.sup.1 to R.sup.6 are each hydrogen, C.sub.1 -C.sub.10 -alkyl, 5-membered to 7-membered cycloalkyl which in turn may carry C.sub.1 -C.sub.10 -alkyl as a substituent, C.sub.6 -C.sub.15 -aryl or arylalkyl, where two adjacent radicals together may furthermore form a cyclic group of 4 to 15 carbon atoms, or Si(R.sup.8).sub.3,R.sup.8 is C.sub.1 -C.