Chebyshev potentials, Fubini–Study metrics, and geometry of the space of Kähler metrics

The Chebyshev potential of a Hermitian metric on an ample line bundle over a projective variety, introduced by Witt Nyström, is a convex function defined on the Okounkov body. It is a generalization of the symplectic potential of a torus-invariant Kähler potential on a toric variety, introduced by Guillemin, that is a convex function on the Delzant polytope. A folklore conjecture asserts that a curve of Chebyshev potentials associated to a subgeodesic in the space of positively curved Hermitian metrics is linear in the time variable if and only if the subgeodesic is a geodesic in the Mabuchi metric. This is classically true in the special toric setting, and in general Witt Nyström established the sufficiency. The main obstacle in the conjecture is that it is difficult to compute Chebyshev potentials, that are currently only known on the Riemann sphere and toric varieties. The goal of this article is to disprove this conjecture. To that end we characterize the geodesics consisting of Fubini–Study metrics for which the conjecture is true on the hyperplane bundle of the projective space. The proof involves explicitly solving the Monge–Ampère equation describing geodesics on the subspace of Fubini–Study metrics and computing their Chebyshev potentials.