Prym varieties and integrable systems
In this article a new relation between Prym varieties of arbitrary morphisms of algebraic curves and integrable systems is discovered. The action of maximal commutative subalgebras of the formal loop algebra of \(GL_n\) defined on certain infinite-dimensional Grassmannians is studied. H. Morikawa's question is: What are the finite-dimensional orbits of these Heisenberg flows, and what kind of geometric objects do they represent? Theorem A. A finite-dimensional orbit of the Heisenberg flow defined on the Grassmannian of vector valued functions corresponds to a covering morphism of algebraic curves, and the orbit itself is canonically isomorphic to the Jacobian variety of the curve upstairs. Moreover, the action of the traceless elements of the Borel subalgebra (the traceless Heisenberg flows) produces the Prym variety associated with this covering morphism as an orbit. Let us define the Grassmannian quotient \(Z_n(0)\) as the quotient space of \(\text{Gr}_n(0)\) by the diagonal action of \((1+\mathbb Czz)^{\times n}.\) The traceless \(n\)-component KP system is defined by the action of the traceless diagonal matrices with entries in \(\mathbb C[z^{-1}]\) on \(Z_n(0)\). Since this system is a special case of the traceless Heisenberg flows, every finite-dimensional orbit of this system is a Prym variety. Conversely, an arbitrary Prym variety associated with a degree n covering morphism of algebraic curves can be realized as finite-dimensional orbit. Theorem B. An algebraic variety is isomorphic to the Prym variety associated with a degree n covering of an algebraic curve if and only if it can be realized as a finite-dimensional orbit of the traceless \(n\)-component KP system defined on the Grassmannian quotient \(Z_n(0)\). Theorem D. Every object of the category \(C(n)\) with a smooth curve \(C_n\) and a line bundle \(F\) on \(C_n\) satisfying the cohomology vanishing condition \(H^0(C_n,F)= H^1(C_n,F)=0\) gives rise to a maximal commutative algebra consisting of ordinary differential operators with coefficients in \(n\times n\) matrix valued functions. Theorem E. The big-cell of the Grassmannian \(\text{Gr}_n(0)\) is canonically identified with the group of monic invertible pseudodifferential operators with matrix coefficients.
algebra of ordinary differential operators
scientific article; zbMATH DE number 1127778