Classification of real Riemann surfaces and their Jacobians in the critical case
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For every g≥ 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.
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