We study when the period and the index of a class in the Brauer group of the function field of a real algebraic surface coincide. We prove that it is always the case if the surface has no real points (more generally, if the class vanishes in restriction to the real points of the locus where it is well-defined), and give a necessary and sufficient condition for unramified classes. As an application, we show that the \(u\)-invariant of the function field of a real algebraic surface is equal to 4, answering questions of Lang and Pfister. Our strategy relies on a new Hodge-theoretic approach to de Jong’s period-index theorem on complex surfaces.

我们研究实数代数曲面的函数域的Brauer组中某个类的周期和索引何时一致。我们证明，如果表面没有实点（在更一般的情况下，如果类在对定义明确的位点的实点的限制中消失），总是如此，并且为未分叉的类提供了必要和充分的条件。作为一个应用，我们证明了真实代数曲面的函数场的\（u \）不变性等于4，回答了Lang和Pfister的问题。我们的策略依靠一种新的Hodge理论方法来处理复杂表面上De Jong的周期指数定理。