A real period-index theorem. (Нестандартное время и аудитория семинара!)

Lang has conjectured that a quadratic form in at least 5 variables over the function field of a real surface with no real points has a nontrivial zero. After explaining geometric motivations for this question, we will sketch a proof relying on a real variant of de Jong’s period-index theorem, and based on Hodge-theoretic arguments.