After a complete hysterectomy will polycystic ovarian syndrome go away?

No. Pcos does not "go away" unless the ovaries are removed at the same time. However, after a hysterectomy, the periods will not be an issue and pcos symptoms may not ba a factor any longer.
Ovaries removed? Only if the ovaries were removed at the time of hysterectomy. I don't want to assume too much here, 'complete hysterectomy' has no medical definition.