In 1988 Atiyah and Hitchin introduced a Poisson bracket (PB) on meromorphic functions defined on the Riemann sphere. Can one replace the Riemann sphere by a Riemann surface of genus g >0? We present one such generalization based on the theory of completely integrable systems.
The periodic problem for the Korteveg de Vriez equation, the cubic nonlinear Schrodinger equation and the like is solved by methods of algebraic geometry. Novikov, Veselov and Dubrovin in 1982 singled out a class of PB for these equations. They call these brackets analytic PB compatible with algebraic geometry. A systematic theory of analytic PB is still lacking.
Based on a notion of Weyl function defined on a Riemann surface we obtain a new formula for the basic PB. This deformed Atiyah--Hitchin PB is a partial answer to the question in the title. We discuss some open problems.