nruddock, englishbob,
I found a solution by rebuilding the base of the stairs but with less splines and vertices. The shady parts didn't  show up anymore. The fact is that I'm a novice to modeling and the first sate of the obj was in some way the result of trail and error. Maybe that caused some bad parts. I don't know if that makes sense to you, but it worked. Thanks anyway for your support!
Regards Herbert