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Motivated by conjectures in Mirror Symmetry, we continue the study of the real Monge--Amp\`ere 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., Comment: 13 pages

We show that the metric defined by the solution to the tropical Monge-Amp\`ere equation, as defined by Hultgren, Mazzon, and the first two authors, on the boundary of the 3-simplex is asymptotic to the Gross-Wilson metric on $S^2$ near each of the 6 singular points. We deduce in addition that the solution is not $C^{1,1}$ across the singular points. Compared to previous works, our starting point is the real Monge-Amp\`ere equation, as opposed to the complex structure., Comment: 16 pages, 4 figures. Comments welcome!

We introduce a synthetic approach to global pluripotential theory, covering in particular the case of a compact K\"ahler manifold and that of a projective Berkovich space over a non-Archimedean field. We define and study the space of measures of finite energy, introduce twisted energy and free energy functionals thereon, and show that coercivity of these functionals is an open condition with respect to the polarization., Comment: 48 pages

We study the non-Archimedean Monge-Amp\`ere equation on a smooth projective variety over a discretely or trivially valued field. First, we give an example of a Green's function, associated to a divisorial valuation, which is not Q-PL (i.e. not a model function in the discretely valued case). Second, we produce an example of a function whose Monge-Amp\`ere measure is a finite atomic measure supported in a dual complex, but which is not invariant under the retraction associated to any snc model. This answers a question by Burgos Gil et al in the negative. Our examples are based on geometric constructions by Cutkosky and Lesieutre, and arise via base change from Green's functions over a trivially valued field; this theory allows us to efficiently encode the Zariski decomposition of a pseudoeffective numerical class., Comment: 39 pages

We give new proofs of two implications in the Donaldson--Uhlenbeck--Yau theorem. Our proofs are based on geodesic rays of Hermitian metrics, inspired by recent work on the Yau--Tian--Donaldson conjecture., Comment: 24 pgs; v2: fixed some notational inconsistencies

For a class of maximally degenerate families of Calabi-Yau hypersurfaces of complex projective space, we study associated non-Archimedean and tropical Monge-Amp\`ere 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., Comment: To appear in Adv. Math

To any projective pair $(X,B)$ equipped with an ample $\mathbb{Q}$-line bundle $L$ (or even any ample numerical class), we attach a new invariant $\beta(\mu)\in\mathbb{R}$, 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., Comment: Final version, to appear in Crelle's journal

This note is an addendum to the paper `Global pluripotential theory over a trivially valued field' by the present authors, in which we prove two results. Let $X$ be an irreducible projective variety over an algebraically closed field field $k$, and assume that $k$ has characteristic zero, or that $X$ has dimension at most two. We first prove that when $X$ is smooth, the envelope property holds for any numerical class on $X$. Then we prove that for $X$ possibly singular and for an ample numerical class, the Monge--Amp\`ere energy of a bounded function is equal to the energy of its usc regularized plurisubharmonic envelope., Comment: 9 pages, addendum to the paper arXiv:1801.08229. To appear in Ann. Fac. Sci. Toulouse Math

For any polarized variety (X,L), we show that test configurations and, more generally, R-test configurations (defined as finitely generated filtrations of the section ring) can be analyzed in terms of Fubini-Study functions on the Berkovich analytification of X with respect to the trivial absolute value on the ground field. Building on non-Archimedean pluripotential theory, we describe the (Hausdorff) completion of the space of test configurations, with respect to two natural pseudo-metrics, in terms of plurisubharmonic functions and measures of finite energy on the Berkovich space. We also describe the Hausdorff quotient of the space of all filtrations, and establish a 1--1 correspondence between divisorial norms and divisorial measures, both being determined in terms of finitely many divisorial valuations., Comment: 70 pages. Final version, to appear in Ann. Inst. Fourier

We give examples of birational selfmaps of $\mathbb{P}^d, d \geq 3$, 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., Comment: To appear in Proc. Lond. Math. Soc

We give an example of a dominant rational selfmap of the projective plane whose dynamical degree is a transcendental number., Comment: 26 pages. Exposition has been changed after receiving a careful referee report. To appear in Acta Math

Let $K$ be a non-archimedean local field, $X$ a smooth and proper $K$-scheme, and fix a pluricanonical form on $X$. For every finite extension $K'$ of $K$, the pluricanonical form induces a measure on the $K'$-analytic manifold $X(K')$. We prove that, when $K'$ runs through all finite tame extensions of $K$, suitable normalizations of the pushforwards of these measures to the Berkovich analytification of $X$ converge to a Lebesgue-type measure on the temperate part of the Kontsevich--Soibelman skeleton, assuming the existence of a strict normal crossings model for $X$. We also prove a similar result for all finite extensions $K'$ under the assumption that $X$ has a log smooth model. This is a non-archimedean counterpart of analogous results for volume forms on degenerating complex Calabi--Yau manifolds by Boucksom and the first-named author. Along the way, we develop a general theory of Lebesgue measures on Berkovich skeleta over discretely valued fields.

We study K-stability properties of a smooth Fano variety X using non-Archimedean geometry, specifically the Berkovich analytification of X with respect to the trivial absolute value on the ground field. More precisely, we view K-semistability and uniform K-stability as conditions on the space of plurisubharmonic (psh) metrics on the anticanonical bundle of X. Using the non-Archimedean Calabi-Yau theorem and the Legendre transform, this allows us to give a new proof that K-stability is equivalent to Ding stability. By choosing suitable psh metrics, we also recover the valuative criterion of K-stability by Fujita and Li. Finally, we study the asymptotic Fubini-Study operator, which associates a psh metric to any graded filtration (or norm) on the anticanonical ring. Our results hold for arbitrary smooth polarized varieties, and suitable adjoint/twisted notions of K-stability and Ding stability. They do not rely on the Minimal Model Program.

We develop global pluripotential theory in the setting of Berkovich geometry over a trivially valued field. Specifically, we define and study functions and measures of finite energy and the non-Archimedean Monge-Ampere operator on any (possibly reducible) projective variety. We also investigate the topology of the space of valuations of linear growth, and the behavior of psh functions thereon., Comment: 143 pages, 1 figure. Various improvements, psh functions now defined and studied relative to any (not necessarily ample) class. Final version, to appear in Ann. Fac. Sci. Toulouse

Let X be a normal complex projective variety with at worst klt singularities, and L a big line bundle on X. We use valuations to study the log canonical threshold of L, as well as another invariant, the stability threshold. The latter generalizes a notion by Fujita and Odaka, and can be used to characterize when a Q-Fano variety is K-semistable or uniformly K-stable. It can also be used to generalize volume bounds due to Fujita and Liu. The two thresholds can be written as infima of certain functionals on the space of valuations on X. When L is ample, we prove that these infima are attained. In the toric case, toric valuations acheive these infima, and we obtain simple expressions for the two thresholds in terms of the moment polytope of L., Comment: minor changes, to appear in Adv. Math

We study the asymptotic behavior of volume forms on a degenerating family of compact complex manifolds. Under rather general conditions, we prove that the volume forms converge in a natural sense to a Lebesgue-type measure on a certain simplicial complex. In particular, this provides a measure-theoretic version of a conjecture by Kontsevich--Soibelman and Gross--Wilson, bearing on maximal degenerations of Calabi--Yau manifolds., Comment: Minor changes. To appear in Journal de l'Ecole polytechnique

Consider a polarized complex manifold (X,L) and a ray of positive metrics on L defined by a positive metric on a test configuration for (X,L). For most of the common functionals in K\"ahler geometry, we prove that the slope at infinity along the ray is given by evaluating the non-Archimedean version of the functional (as defined in our earlier paper) at the non-Archimedean metric on L defined by the test configuration. Using this asymptotic result, we show that coercivity of the Mabuchi functional implies uniform K-stability., Comment: New version with errata (an error was found in the proof of Theorem 5.6). The affected parts of the paper are marked in red

We give a variational proof of a version of the Yau-Tian-Donaldson conjecture for twisted K\"ahler-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 the continuity method or 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., Comment: Added Appendix B on a valuative analysis of singularities of plurisubharmonic functions. Various other small changes and improvements. To appear in Journal of the AMS

We show that any birational selfmap of a complex projective surface that has dynamical degree greater than one and is defined over a number field automatically satisfies the Bedford-Diller energy condition after a suitable birational conjugacy. As a consequence, the complex dynamics of the map is well-behaved. We also show that there is a well-defined canonical height function., Comment: Added simplified argument for algebraically stable maps on the projective plane. To appear in Crelle's Journal. 19 pages

The purpose of the present paper is to set up a formalism inspired from non-Archimedean geometry to study K-stability. We first provide a detailed analysis of Duistermaat-Heckman measures in the context of test configurations, characterizing in particular the trivial case. For any normal polarized variety (or, more generally, polarized pair in the sense of the Minimal Model Program), we introduce and study the non-Archimedean analogues of certain classical functionals in K\"ahler geometry. These functionals are defined on the space of test configurations, and the Donaldson-Futaki invariant is in particular interpreted as the non-Archimedean version of the Mabuchi functional, up to an explicit error term. Finally, we study in detail the relation between uniform K-stability and singularities of pairs, reproving and strengthening Y. Odaka's results in our formalism. This provides various examples of uniformly K-stable varieties., Comment: Small changes. To appear in Ann. Inst. Fourier

We give an introduction to our work on the solution to the non-Archimedean Monge-Ampere equation and make comparisons to the complex counterpart. These notes are partially based on talks at the 2015 Simons Symposium on Tropical and Nonarchimedean Geometry., Comment: To appear in the Proceedings of the Simons Symposia on Tropical and Non-Archimedean Geometry

We prove a continuity result for the fibers of the Berkovich analytification of a complex algebraic variety with respect to the the maximum of the Archimedean norm and the trivial norm. As a consequence, we obtain generalizations of a result of Mikhalkin and Rullgard about degenerations of amoebae onto tropical varieties., Comment: To appear in Math. Ann

We improve Izumi's inequality, which states that any divisorial valuation v centered at a closed point 0 on an algebraic variety Y is controlled by the order of vanishing at 0. More precisely, as v ranges through valuations that are monomial with respect to coordinates in a fixed birational model X dominating Y, we show that for any regular function f on Y at 0, the function v--> v(f)/\ord_0(f) is uniformly Lipschitz continuous as a function of the weight defining v. As a consequence, the volume of v is also a Lipschitz continuous function. Our proof uses toroidal techniques as well as positivity properties of the images of suitable nef divisors under birational morphisms., Comment: 23 pages, 1 figure

We reduce the Openness Conjecture of Demailly and Koll\'ar on the singularities of plurisubharmonic functions to a purely algebraic statement., Comment: Small modifications. 26 pages. To appear in J. Inst. Math. Jussieu

We study canonical heights for plane polynomial mappings of small topological degree. In particular, we prove that for points of canonical height zero, the arithmetic degree is bounded by the topological degree and hence strictly smaller than the first dynamical degree. The proof uses the existence, proved by Favre and the first author, of certain compactifications of the plane adapted to the dynamics., Comment: Modified title and some references in response to the new version of the paper by Silverman. 10 pages. To appear in Math. Res. Lett