Integrable Systems in the Realm of Algebraic Geometry by Pol Vanhaecke PDF

Algebraic Geometry

By Pol Vanhaecke

This ebook treats the final conception of Poisson buildings and integrable structures on affine kinds in a scientific means. particular cognizance is attracted to algebraic thoroughly integrable platforms. numerous integrable structures are developed and studied intimately and some functions of integrable structures to algebraic geometry are labored out. within the moment version a number of the techniques in Poisson geometry are clarified by way of introducting Poisson cohomology; the Mumford platforms are made from the algebra of pseudo-differential operators, which clarifies their beginning; a brand new rationalization of the multi Hamiltonian constitution of the Mumford structures is given through the use of the loop algebra of sl(2); and at last Goedesic circulation on SO(4) is additional to demonstrate the linearizatin algorith and to provide one other software of integrable structures to algebraic geometry.

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The algebraic setup which we use here has the virtue to allow to pass easily to the quotient (one does so not need to worry about the action being free, picking regular values and on). 25 Let G be a finite or reductive group and consider a Poisson action M, where (M, 1-, -1) is an affine Poisson variety. 10, AG) is an involutive Hamiltonian system 9 and the quotient map -7r is a morphism. 25. }O, AG) is integrable. 25. Suppose now that G is finite. completeness of A implies completeness of A n O(M)G.

Integrable Hamiltonian systems and multi-Hamiltonian introduce a integrable systems few concepts which relate to compatible integrable Hamiltonian systems. 30 brackets Let affine i variety M. =-= If 1, n be (linearly independent) compatible n (M, I-, ji, A) is Poisson integrable Hamiltonian system for each i n then these systems axe called compatible integrable Hamiltonian 1, systems. , for which there exist fl, f,, E A such that on an an = . . , . , fill = = ... (bi-Hamiltonian many different ways; any of the an = .

A =A 0 and look for a Jp1 P21 linear function G alql + b'q2 + 41 + dIP2 which is in involution with F. Replacing G by = = I I i = = = 7 = G - Fa'/a if necessary we may assume G = b1q2 that a' + (db' == - 0 and bd)pl we + find dP2 general solution (up to adding multiples of F). Here Y, d' EE C are arbitrary, so essentially a one-paxameter family of possibilities for G (paxametrized by d1b'), The Poisson bracket of two of these all leading to an integrable subalgebra A of O(C4) for is by G, given possibilities as the most that we have .

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