Sixth Painlevé equation, universal elliptic curve, and mirror of \(\mathbb{P}^2\). (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 \[ \begin{split}\frac{dX}{dt^2}=\frac{1}{2} \Biggl( \frac{1}{X}+\frac{1}{X-1}+\frac{1}{X-t} \Biggr) \Biggl(\frac{dX}{dt} \Biggr)^2- \Biggl( \frac{1}{t}+\frac{1}{t-1}+\frac{1}{X-t} \Biggr)\frac{dX}{dt}\\ +\frac{X(X-1)(X-t)}{t^2(t-1)^2} \Biggl( \alpha+\beta\frac{t}{X^2}+\gamma\frac{t-1} {(X-1)^2}+\delta\frac{t(t-1)}{(X-t)^2}\Biggr), \end{split} \] 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.
For the entire collection see [Zbl 0896.00014].


14H52 Elliptic curves
34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
34M15 Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical) of ordinary differential equations in the complex domain
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14K20 Analytic theory of abelian varieties; abelian integrals and differentials
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