Algebraic de Rham theorem and Baker--Akhiezer function

I. M. Krichever ; L. A. Takhtadzhyan

Izvestiya. Mathematics , Tome 88 (2024) no. 3, pp. 506-514

Résumé

For the case of algebraic curves (compact Riemann surfaces), it is shown that de Rham cohomology group $H^1_{\mathrm{dR}}(X,\mathbb{C})$ of a genus $g$ of the 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 $H^1_{\mathrm{dR}}(X,\mathbb{C})$ consisting of holomorphic differentials and differentials of the second kind with poles on $D$. This result, which is 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 the curve $X$ 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.

Détail

Keywords: Riemann surface, divisor, line bundle, Riemann–Roch theorem, differentials of the second kind, algebraic de Rham theorem, Picard and Jacobian varieties, vector field on the Jacobian variety, Lax representation, Dubrovin equation, Baker–Akhiezer function.