2023/11/08 by I. M. Krichever, Leon A. Takhtajan, Krichever, Igor +1
Mathematics · #14F40 #14H40 #14H70 #Advanced Algebra and Geometry #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2311.04440
openalex publication_date 2023/11/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For the case of algebraic curves - compact Riemann surfaces - it is shown that de Rham cohomology group H1dR(X,ℂ) of a genus g Riemann surface X has a natural structure of a symplectic vector space. Every choice of a non-special effective divisor D of degree g on X defines a symplectic basis of H1dR(X,ℂ), consisting of holomorphic differentials and differentials of the second kind with poles on D. This result, the algebraic de Rham theorem, is used to describe the tangent space to Picard and Jacobian varieties of X in terms of differentials of the second kind, and to define a natural vector fields on the Jacobian of X that move points of the divisor D. In terms of the Lax formalism on algebraic curves, these vector fields correspond to the Dubrovin equations in the theory of integrable systems, and the Baker-Akhierzer function is naturally obtained by the integration along the integral curves.