To provide an olefin polymerization catalyst having high catalytic activity (polymerization activity), an olefin polymerization catalyst production method, and an olefin polymer production method using the olefin polymerization catalyst. An olefin polymerization catalyst comprising the following components [A], [B] and [C]: the component [A]: a metallocene complex having a specific indenyl structure, the component [B]: a solid component containing one or more selected from the group consisting of (b-1) a fine particulate carrier on which an ionic compound or Lewis acid which can convert the component [A] into a cation by reaction with the component [A], is supported, (b-2) solid acid fine particles, and (b-3) an ion-exchange layered compound, and the component [C]: a specific silylacetylene compound.