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Contemporary Mathematics 

Abstract Motivated by conjectures in Mirror Symmetry, we continue the study of the real Monge–Ampère operator on the boundary of a simplex. This can be formulated in terms of optimal transport, and we consider, more generally, the problem of optimal transport between symmetric probability measures on the boundary of a simplex and of the dual simplex. For suitably regular measures, we obtain regularity properties of the transport map, and of its convex potential. To do so, we exploit boundary regularity results for optimal transport maps by Caffarelli, together with the symmetries of the simplex. 

For a large class of maximally degenerate families of Calabi–Yau hypersurfaces of complex projective space, we study non- Archimedean and tropical Monge–Ampère equations, taking place on the associated Berkovich space, and the essential skeleton therein, respectively. For a symmetric measure on the skeleton, we prove that the tropical equation admits a unique solution, up to an additive constant. Moreover, the solution to the non-Archimedean equation can be derived from the tropical solution, and is the restriction of a continuous semipositive toric metric on projective space. Together with the work of Yang Li, this implies the weak metric SYZ conjecture on the existence of special Lagrangian fibrations in our setting. 

To any projective pair (X,B) equipped with an ample Q-line bundle L (or even any ample numerical class), we attach a new invariant $$\beta(\mu)$$, defined on convex combinations $$\mu$$ of divisorial valuations on X , viewed as point masses on the Berkovich analytification of X . The construction is based on non-Archimedean pluripotential theory, and extends the Dervan–Legendre invariant for a single valuation – itself specializing to Li and Fujita’s valuative invariant in the Fano case, which detects K-stability. Using our $$\beta$$-invariant, we define divisorial (semi)stability, and show that divisorial semistability implies (X,B) is sublc (i.e. its log discrepancy function is non-negative), and that divisorial stability is an open condition with respect to the polarization L. We also show that divisorial stability implies uniform K-stability in the usual sense of (ample) test configurations, and that it is equivalent to uniform K-stability with respect to all norms/filtrations on the section ring of (X,L), as considered by Chi Li. 

Abstract We give examples of birational selfmaps of , whose dynamical degree is a transcendental number. This contradicts a conjecture by Bellon and Viallet. The proof uses a combination of techniques from algebraic dynamics and diophantine approximation. 

We give a variational proof of a version of the Yau–Tian–Donaldson conjecture for twisted Kähler–Einstein currents, and use this to express the greatest (twisted) Ricci lower bound in terms of a purely algebro-geometric stability threshold. Our approach does not involve a continuity method or the Cheeger–Colding–Tian theory, and uses instead pluripotential theory and valuations. Along the way, we study the relationship between geodesic rays and non-Archimedean metrics.