Sixth Painlevé equation, universal elliptic curve, and mirror of ${\mathbb{P}}^{2}$.

*(English)*Zbl 0948.14025Khovanskij, 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

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.

Reviewer: Lucian Bădescu (Los Angeles)