Prym varieties and integrable systems
type: Item
instance of: scholarly article
title: Prym varieties and integrable systems
zbMATH Open document ID: 0920.14010
author: Yingchen Li
author: Motohico Mulase
published in: Communications in Analysis and Geometry
publication date: 1998-05-13T00:00:00Z
review text: 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 …
Mathematics Subject Classification ID: 14H40
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 14E20
Mathematics Subject Classification ID: 14E22
Mathematics Subject Classification ID: 14M15
zbMATH DE Number: 1127778
zbMATH Keywords: Prym varieties
zbMATH Keywords: integrable systems
zbMATH Keywords: morphisms of algebraic curves
zbMATH Keywords: formal loop algebra
zbMATH Keywords: infinite-dimensional Grassmannians
zbMATH Keywords: Heisenberg flows
zbMATH Keywords: traceless \(n\)-component KP system
zbMATH Keywords: cohomology vanishing condition
zbMATH Keywords: algebra of ordinary differential operators
reviewed by: V. V. Chueshev
MaRDI profile type: Publication
full work available at URL: cag.1997.v5.n2.a3
OpenAlex ID: W1620259270
DOI: 10.4310/CAG.1997.V5.N2.A3
description: scientific article; zbMATH DE number 1127778