Algebraic de Rham theorem and Baker–Akhiezer function

For the case of algebraic curves (compact Riemann surfaces), it is shown thatde 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 $H^1_{\mathrm{dR}}(X,\mathbb{C})$ consistingof holomorphic differentials and differentials of the second kind with poleson $D$. This result, which is the algebraic de Rham theorem, is used to describethe tangent space to Picard and Jacobian varieties of $X$in terms of differentials of the second kind, and to define a naturalvector 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 fieldscorrespond to the Dubrovin equations in the theory of integrable systems,and the Baker–Akhierzer function is naturally obtained by the integration alongthe integral curves.

Riemann surface, divisor, line bundle, Riemann–Roch theorem, differentials of the second kind, algebraic de Rham theorem, Picardand Jacobian varieties, vector field on the Jacobian variety, Laxrepresentation, Dubrovin equation, Baker–Akhiezer function