Numerical Kaehler-Einstein metric on the third del Pezzo

TL;DR

We numerically construct the unique Kähler-Einstein metric on the toric surface $dP_3$ by reducing the Einstein equations to a Monge-Ampère equation in two real variables and solving it with three complementary methods: Ricci flow in complex coordinates, Ricci flow in symplectic coordinates, and a constrained optimization using symplectic polynomials. The three approaches yield consistent metrics at the $10^{-6}$ level and enable computation of geometric quantities such as Laplacian eigenvalues and harmonic $(1,1)$-forms, which feed into a Klebanov-Tseytlin-like flux background and gauge/gravity duality constructions. The work demonstrates the applicability of toric methods to non-Calabi–Yau settings, provides compact analytic fits via invariant polynomials, and offers data and notebooks for broader use in geometry and physics. It also outlines natural extensions to other toric Fano manifolds and possible higher-dimensional generalizations, including smooth Calabi–Yau cones over $dP_3$ for broader holographic applications.

Abstract

The third del Pezzo surface admits a unique Kaehler-Einstein metric, which is not known in closed form. The manifold's toric structure reduces the Einstein equation to a single Monge-Ampere equation in two real dimensions. We numerically solve this nonlinear PDE using three different algorithms, and describe the resulting metric. The first two algorithms involve simulation of Ricci flow, in complex and symplectic coordinates respectively. The third algorithm involves turning the PDE into an optimization problem on a certain space of metrics, which are symplectic analogues of the "algebraic" metrics used in numerical work on Calabi-Yau manifolds. Our algorithms should be applicable to general toric manifolds. Using our metric, we compute various geometric quantities of interest, including Laplacian eigenvalues and a harmonic (1,1)-form. The metric and (1,1)-form can be used to construct a Klebanov-Tseytlin-like supergravity solution.