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Zbl 0948.14025
Manin, Yu.I.
Sixth Painlevé equation, universal elliptic curve, and mirror of $\bbfP^2$. (English)
[A] 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). ISBN 0-8218-1094-4/hbk

The sixth Painlevé equation studied geometrically in this paper is the following $$\multline\frac{dX}{dt^2}=\frac{1}{2} \Biggl( \frac{1}{X}+\frac{1}{X-1}+\frac{1}{X-t} \Biggr) \Biggl(\frac{dX}{dt} \Biggr)^2- \Biggl( \frac{1}{t}+\frac{1}{t-1}+\frac{1}{X-t} \Biggr)\frac{dX}{dt}\\ +\frac{X(X-1)(X-t)}{t^2(t-1)^2} \Biggl( \alpha+\beta\frac{t}{X^2}+\gamma\frac{t-1} {(X-1)^2}+\delta\frac{t(t-1)}{(X-t)^2}\Biggr), \endmultline$$ where $\alpha$, $\beta$, $\gamma$ and $\delta$ 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.
[Lucian Bădescu (Los Angeles)]

MSC 2000:: *14H52 Elliptic curves
34M55 Painlevé and other special equations
34M15 Algebraic aspects of differential equations in the complex domain
14N35 Quantum cohomology
14K20 Analytic theory; abelian integrals and differentials

Keywords: Painlevé equation; universal elliptic curve; abelian integrals; quantum cohomology

Cited in: Zbl 1146.32005 Zbl 1097.32007 Zbl 1081.14503

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