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Sixth Painlevé equation, universal elliptic curve, and mirror of 2 . (English) Zbl 0948.14025
Khovanskij, A. (ed.) et al., Geometry of differential equations. Dedicated to V. I. Arnold on the occasion of his 60th birthday. Providence, RI: American Mathematical Society. Transl., Ser. 2, Am. Math. Soc. 186(39), 131-151 (1998).

The sixth Painlevé equation studied geometrically in this paper is the following

dX dt 2 =1 21 X + 1 X-1 + 1 X-tdX dt 2 -1 t + 1 t-1 + 1 X-tdX dt+X(X-1)(X-t) t 2 (t-1) 2 α + β t X 2 + γ t-1 (X-1) 2 + δ t(t-1) (X-t) 2 ,

where α, β, γ and δ are four parameters. This equation has been studied classically from several points of view. In the paper under review the author takes up a new approach via abelian integrals and algebraic geometry. The first thing he does is to introduce an algebro-geometric setting for this equation. Then a Hamiltonian form of the equation is realized on a twisted relative cotangent bundle to the universal elliptic curve with labeled points of order two. Finally, some relation with the theory of quantum cohomology of projective plane is discussed.


MSC:
14H52Elliptic curves
34M55Painlevé and other special equations; classification, hierarchies
34M15Algebraic aspects of ODE in the complex domain
14N35Gromov-Witten invariants, quantum cohomology, etc.
14K20Analytic theory; abelian integrals and differentials