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Symplectic manifolds and isomonodromic deformations. (English) Zbl 1001.53059
The author studies the moduli spaces of meromorphic connections with poles of arbitrary order over Riemann surfaces together with the corresponding spaces of monodromy data involving Stokes matrices. Generalizing the Atiyah-Bott approach [M. F. Atiyah and R. Bott, Phil. Trans. R. Soc. Lond. 308, 523-615 (1983; Zbl 0509.14014)], natural symplectic structures on extended spaces of singular connections are found and described both explicitly and from an infinite dimensional viewpoint. By the symplectic space construction an intrinsic symplectic description of the isomonodromic deformation equations of Jimbo, Miwa and Ueno is given and the existing results for the six Painlevé equations and Schlesinger’s equations are put into a uniform framework.

MSC:
53D20 Momentum maps; symplectic reduction
34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
53D30 Symplectic structures of moduli spaces
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