On the integrability of symplectic Monge-Ampere equations

Let u be a function of n independent variables x^1, …, x^n, and U=(u_{ij}) the Hessian matrix of u. The symplectic Monge-Ampere equation is defined as a linear relation among all possible minors of U. Particular examples include the equation det U=1 governing improper affine spheres and the so-called heavenly equation, u_{13}u_{24}-u_{23}u_{14}=1, describing self-dual Ricci-flat 4-manifolds. In this paper we classify integrable symplectic Monge-Ampere equations in four dimensions (for n=3 the integrability of such equations is known to be equivalent to their linearisability). This problem can be reformulated geometrically as the classification of ‘maximally singular’ hyperplane sections of the Plucker embedding of the Lagrangian Grassmannian. We formulate a conjecture that any integrable equation of the form F(u_{ij})=0 in more than three dimensions is necessarily of the symplectic Monge-Ampere type.

💡 Research Summary

The paper undertakes a systematic classification of integrable symplectic Monge‑Ampère equations in four independent variables. A symplectic Monge‑Ampère equation is defined as a linear relation among all minors of the Hessian matrix (U=(u_{ij})) of a scalar function (u(x^1,\dots ,x^n)). In four dimensions the Hessian is a (4\times4) symmetric matrix, and the possible minors range from first‑order entries to the full determinant. The authors recast the problem geometrically: the Lagrangian Grassmannian (\Lambda(4)) of 4‑dimensional Lagrangian subspaces is embedded into projective space (\mathbb{P}^9) via the Plücker map, where each point corresponds to the collection of all minors of a symmetric matrix. A hyperplane section of this embedding corresponds precisely to a linear relation among the minors, i.e. a symplectic Monge‑Ampère equation.

The key geometric notion introduced is that of a “maximally singular” hyperplane – a hyperplane that is tangent to (\Lambda(4)) to the highest possible order. Such hyperplanes generate the most degenerate intersections and therefore the most restrictive linear relations among the minors. By analysing the tangent variety of (\Lambda(4)) the authors distinguish two qualitatively different tangency patterns. In the first, the hyperplane meets the tangent variety transversally; the resulting equation can be written as a product of two second‑order forms. The prototypical example is the heavenly equation \