A piano fingering indicates which finger should play each note in a piece. Such a guideline is very helpful for both amateur and experienced players in order to play a piece fluently. In this paper, we propose a variable neighborhood search algorithm to generate piano fingerings for complex polyphonic music, a frequently encountered case that was ignored in previous research. The algorithm takes into account the biomechanical properties of the pianist's hand in order to generate a fingering that is user-specific and as easy to play as possible. An extensive statistical analysis was carried out in order to tune the parameters of the algorithm and evaluate its performance. The results of computational experiments show that the algorithm generates good fingerings that are very similar to those published in sheet music books.