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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.

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