Integrable linear equations and the Riemann-Schottky problem
We prove that an indecomposable principally polarized abelian variety X is the Jacobain of a curve if and only if there exist vectors U ≠ 0, V such that the roots x i(y) of the theta-functional equation θ(Ux + Vy + Z) = 0 satisfy the equations of motion of the formal infinite-dimensional Calogero-Moser system.
KeywordsWave Solution Meromorphic Function Abelian Variety Pseudodifferential Operator Spectral Curve
Unable to display preview. Download preview PDF.
- V. S. Druma, On analytic solution of the two-dimensional Korteweg-de Vries equation, JETP Lett., 19-12 (1974), 219–225.Google Scholar
- E. Arbarello, I. Krichever, and G. Marini, Characterizing Jacobians via flexes of the Kummer variety, 2005, math.AG/0502138.Google Scholar
- I. Krichever and D. H. Phong, Symplectic forms in the theory of solitons, in C. L. Terng and K. Uhlenbeck, eds., Surveys in Differential Geometry IV, International Press, Cambridge, MA, 1998, 239–313.Google Scholar
- D. Mumford, An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg-de Vries equation and related non-linear equations, in Proceedings of the International Symposium on Algebraic Geometry, Kyoto, 1977, Kinokuniya Book Store, Kyoto, 1978, 115–153.Google Scholar
- G. Sigal and G. Wilson, Loop groups and equations of KdV type, IHES Publ. Math., 61 (1985), 5–65.Google Scholar