For every $g\geq 2$ we distinguish real period matrices of real Riemann surfaces of topological type $(g,0,0)$ from the ones of topological type $(g,k,1)$, with $k$ equal to one or two for $g$ even or odd respectively (Theorem B). To that purpose, we exhibit new invariants of real principally polarized abelian varieties of orthosymmetric type (Theorem A.1). As a direct application, we obtain an exhaustive criterion to decide about the existence of real points on a real Riemann surface, requiring only a real period matrix of its and the evaluation of the sign of at most one (real) theta constant (Theorem C). A part of our real, algebro-geometric instruments first appeared in the framework of nonlinear integrable partial differential equations.

翻译：对于每个$g\geq 2$，我们区分了拓扑类型为$(g,0,0)$的实Riemann曲面与拓扑类型为$(g,k,1)$的实Riemann曲面的实周期矩阵，其中当$g$为偶数时$k$等于1，当$g$为奇数时$k$等于2（定理B）。为此，我们揭示了正交对称型实主极化Abel簇的新不变量（定理A.1）。作为直接应用，我们获得了判断实Riemann曲面是否存在实点的完备准则，该准则仅需其一个实周期矩阵及至多一个（实）theta常数的符号评估（定理C）。部分实代数几何工具首次出现于非线性可积偏微分方程框架中。