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arXiv · 2307.10486

Classification of real Riemann surfaces and their Jacobians in the critical case

Pietro Giavedoni
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Abstract

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.

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Pietro Giavedoni. 2023-07-19. Classification of real Riemann surfaces and their Jacobians in the critical case. https://arxiv.org/abs/2307.10486

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