On the Hessian Conjecture in Lorentzian Signature: Constant Pivots and Hesse Systems

Abstract

As a close relative of the Jacobian conjecture, the Hessian conjecture in dimension $n$ states that the local Legendre transform of a polynomial solution to the Monge--Ampère equation $\det (\mathrm{Hess}(\varphi ))=\pm 1$ is also a polynomial solution. The general Hessian conjecture is false for $n\ge 5$, while in Riemannian signature it follows from the Jörgens--Calabi--Pogorelov theorem. We study the four-dimensional Hessian conjecture in Lorentzian signature. For a polynomial potential $\varphi$ in four real variables whose Hessian matrix has index $1$ and determinant $-1$, we define a constant pivot for $\varphi$ to be a nonzero constant vector $\xi$ such that the second directional derivative $D_{\xi }^2\varphi$ is constant. We then prove that the gradient mapping of every potential admitting a pivot is a polynomial automorphism, and that a pivot always exists when $\varphi$ decomposes into homogeneous pieces as $\varphi ={\varphi }_d+{\varphi }_{d-1}+{\varphi }_2+{\varphi }_1+{\varphi }_0$ with $d\ge 4$. More generally, we prove the same conclusion when $$\varphi ={\varphi }_d+\cdots +{\varphi }_{d-k}+{\varphi }_2+{\varphi }_1+{\varphi }_0,$$ where $k\ge 0$ and $d\ge 4k+3$. Then, we associate with each potential a linear system of quadrics, called the Hesse system, and a canonical homomorphism ${\mu }_{\varphi }$. We prove that the existence of a pivot is equivalent to $\mathrm{rank}({\mu }_{\varphi })\le 55$. As an application, we prove that the gradient mapping is a polynomial automorphism whenever the Hesse system has complex dimension at most 4. We also show that if $\det (\mathrm{Hess}(\varphi -{\varphi }_2))\equiv 0$, then the potential $\varphi$ admits a pivot. After that, we then give an analytic degeneracy criterion for $\mathrm{Hess}(\varphi -{\varphi }_2)$. Finally, we prove the Hessian conjecture in this setting for every polynomial potential of degree at most five.

arXiv comment57 pages, 0 figure