Seminario Vidal Abascal

Complex projective space ℂPⁿ, which parametrizes the complex lines in ℂⁿ⁺¹, is one of the most extensively studied objects in algebraic and complex geometry. The first Einstein metric discovered on ℂPⁿ was the Fubini–Study metric, and in 1965 Berger asked whether it was the only one. In 1982, Ziller constructed a second homogeneous Einstein metric in odd complex dimensions. Berger's question remained open in even complex dimensions, where Fubini–Study was the only known Einstein metric.

Together with Gonzalo Cao Labora (EPFL, Switzerland), we have constructed new Einstein metrics on ℂPⁿ for n ∈ {3, 4, 5, 6, 7}. These are the first known non-homogeneous examples and, together with the Fubini–Study and Ziller metrics, the only ones known to date. Our construction settles Berger's question in the previously open cases n = 4 and n = 6.

The construction relies on a cohomogeneity-one action by the Lie group SO_n+1, which allows us to reduce the Einstein equations to a system of ODEs, reparametrized by the mean curvature of the principal orbits. One of the novel aspects of the work is the use of a computer-assisted proof, a methodology that has so far been relatively unexplored in differential geometry. We use interval arithmetic and Krawczyk's fixed-point theorem, a generalization of Newton's method that makes it possible to turn numerical analysis into a rigorous proof.

Nearly G₂ manifolds are positive Einstein seven-dimensional manifolds whose associated cone metric has special holonomy equal to, or contained in, Spin(7). We study nearly parallel G₂-structures invariant under a cohomogeneity-one action of SO(4), and construct the first inhomogeneous nearly parallel G₂-structure on the Berger space SO(5)/SO(3). The cone over this metric has full holonomy Spin(7). The existence of the solution is established by a rigorous computer-assisted shooting argument.

We also establish a local desingularisation of a finite quotient of the sine cone over S³ × S³ by gluing in the asymptotically conical C₇ bubble constructed by Foscolo–Haskins–Nordström. We show that as the singular orbit of a family of nearly parallel G₂ manifolds collapses, away from the singular orbit the desingularising family converges to the sine cone quotient, while the natural blow-up converges to the AC torsion-free G₂-metric of C₇. The talk is based on a joint work with Simon Salamon.