Algebraic de Rham theorem and Baker-Akhiezer function (Q6568711)

This short paper provides explicit descriptions of vector space key in the spectral curve approach to integrable systems in terms of function theory.\N\NAs motivation it makes use of a result of Grothendieck which connects the de Rham cohomology \(H^1_{dR}(X, \mathbb{C})\) of a smooth algebraic variety \(X\) to the space of 1-forms of the second kind, that is the closed meromorphic 1-forms which have zero residues outside some sufficiently-large divisor \(D\), denoted \(\Omega^{(2nd)}\). The explicit isomorphism is\N\[\NH^1_{dR}(X, \mathbb{C}) \cong {\Omega^{(2nd)}}/{d\mathcal{M}},\N\]\Nwhere \(\mathcal{M}\) denote the vector space of meromorphic functions on \(X\). This is briefly re-derived at the beginning of \S3.\N\NThis result is considered for genus-\(g\) Riemann surfaces, where \(\Omega^{(2nd)}\) is given the natural skew-symmetric bilinear derived from the reciprocity law. Theorem 1 in \S3 uses elementary properties of divisors and Riemann-Roch to prove that this bilinear is non-degenerate on the quotient, and for each \(D\) a degree-\(g\) non-special effective divisor constructs an implicit symplectic basis of the quotient. In doing so the isomorphism\N\[\N{\Omega^{(2nd)}}/{d\mathcal{M}} \cong \Omega^{(2nd)} \cap H^0(X, K_X+2D)\N\]\Nis shown. \S4 then uses Theorem 1 to express the tangent space to the Picard variety, identified with \(H^{0,1}(X, \mathbb{C})\), as \(\Omega^{(2nd)}(2D)\).\N\N\S5 combines Theorem 1 with the Abel-Jacobi map to describe the tangent space to the point which is the Abel image of \(D\) in terms of the symplectic basis defined by \(D\). This allows for a clean description of integral curves in the Jacobian.