arXiv subjects

Robert J. Berman

Publications and source records attributed to Robert J. Berman.

At least 19 recordsLinked to original sources

arXiv2025

Gibbs polystability of Fano manifolds, stability thresholds and symmetry breaking

We extend the probabilistic approach for constructing Kahler-Einstein metrics on log Fano manifolds X - involving random point processes - to the case of non-discrete automorphism groups, by breaking the symmetry using a moment map constraint. In particular, an algebraic notion of Gibbs polystability is introduced, ensuring that the corresponding point processes on X are well-defined. We conjecture that the Gibbs polystability of X is equivalent to the existence of a Kahler-Einstein metric and that the unique such metric with vanishing moment emerges when sampling a large number of N points on X. The definition of Gibbs polystability involves a limit of log canonical thresholds on the GIT semistable locus of the N-fold products of X, that we conjecture coincides - as N tends to infinity - with an analytic reduced stability threshold, encoding the coercivity of the K-energy functional modulo automorphisms. These conjectures follow from an overarching conjectural Large Deviation Principle for the large N-limit. We prove several of our conjectures on log Fano curves and derive a strengthened form of the sharp logarithmic Hardy-Littlewood-Sobolev (HLS) inequality on the two-sphere, under a moment constraint. It yields quantitative stability results for the sharp logarithmic HLS inequality with optimal stability constants. Furthermore, we show that any log Fano manifold that is strongly uniformly Gibbs polystable admits a Kahler-Einstein metric. In companion papers we will present applications to Onsager's point vortex model on the two-sphere and the AdS/CFT correspondence.

math.DG↗

arXiv2025

Convexity of the K-energy and Uniqueness of Extremal metrics -- An Expository Introduction

This article is an expository introduction to our paper Convexity of the K-energy and Uniqueness of Extremal metrics. We present the main ideas behind the proof that Mabuchi's K-energy functional is convex along weak geodesics in the space of Kahler potentials and explain how this leads to the uniqueness of constant scalar curvature Kahler metrics and extremal metrics up to automorphisms. The emphasis is on the conceptual framework and key techniques.

math.DG↗

arXiv2025

Kähler-Einstein metrics arising from micro-canonical measures and Hamiltonian dynamics

We introduce new probabilistic and variational constructions of (twisted) Kähler-Einstein metrics on complex projective algebraic varieties, drawing inspiration from Onsager's statistical mechanical model of turbulence in two-dimensional incompressible fluids. The probabilistic construction involves microcanonical measures associated with the level sets of the pluricomplex energy, which give rise to maximum entropy principles. These, in turn, yield novel characterizations of Kähler-Einstein metrics, as well of Fano varieties admitting such metrics. Additionally, connections to Hamiltonian dynamics are uncovered, resulting in a new evolution equation, that generalizes both the 2D incompressible Euler equation and the 2D semi-geostrophic equation.

math.DG↗

arXiv2024

Canonical heights, periods and the Hurwitz zeta function

Let (X,D) be a projective log pair over the ring of integers of a number field such that the log canonical line bundle K_(X,D) or its dual -K_(X,D) is relatively ample. We introduce a canonical height of K_(X,D) (and -K(X,D)) which is finite precisely when the complexifications of K_(X,D) (and -K(X,D)) are K-semistable. When the complexifications are K-polystable, the canonical height is the height of K_(X,D) (and -K(X,D)) wrt any volume-normalized Kähler-Einstein metric on the complexifications of K_(X,D) (and -K(X,D)) The canonical height is shown to have a number of useful variational properties. Moreover, it may be expressed as a limit of periods on the N-fold products of the complexifications of X, as N tends to infinity. In particular, using this limit formula, the canonical height for the arithmetic log surfaces (P_1,D) over the integers, where D has at most three components, is computed explicitly in terms of the Hurwitz zeta function and its derivative at s=-1. Combining this explicit formula with previous height formulas for quaternionic Shimura curves yields a procedure for extracting information about the canonical integral models of some Shimura curves, such as wild ramification. Furthermore, explicit formulas for the canonical height of twisted Fermat curves are obtained, implying explicit Parshin type bounds for the Arakelov metric.

math.NT↗

arXiv2024

Reverse Hölder inequalities on the space of Kähler metrics of a Fano variety and effective openness

A reverse Hölder inequality is established on the space of Kähler metrics in the first Chern class of a Fano manifold X endowed with Darvas L^{p}-Finsler metrics. The inequality holds under a uniform bound on a twisted Ricci potential and extends to Fano varieties with log terminal singularities. Its proof leverages a "hidden" log-concavity. An application to destabilizing geodesic rays is provided, which yields a reverse Hölder inequality for the speed of the geodesic. In the case of Aubin's continuity path on a K-unstable Fano variety, the constant in the corresponding Hölder bound is shown to only depend on p and the dimension of X. This leads to some intruiging relations to Harnack bounds and the partial C^{0}-estimate. In another direction, universal effective openness results are established for the complex singularity exponents (log canonical thresholds) of ω-plurisubharmonic functions on any Fano variety. Finally, another application to K-unstable Fano varieties is given, involving Archimedean Igusa zeta functions.

math.DG↗

arXiv2024

Sharp bounds on the height of K-semistable Fano varieties I, the toric case

Inspired by Fujita's algebro-geometric result that complex projective space has maximal degree among all K-semistable complex Fano varieties, we conjecture that the height of a K-semistable metrized arithmetic Fano variety X of relative dimension n is maximal when X is the projective space over the integers, endowed with the Fubini-Study metric. Our main result establishes the conjecture for the canonical integral model of a toric Fano variety when n is less than or equal to 6 (the extension to higher dimensions is conditioned on a conjectural "gap hypothesis" for the degree). Translated into toric Kähler geometry this result yields a sharp lower bound on a toric invariant introduced by Donaldson, defined as the minimum of the toric Mabuchi functional. We furthermore reformulate our conjecture as an optimal lower bound on Odaka's modular height. In any dimension n it is shown how to control the height of the canonical toric model X, with respect to the Kähler-Einstein metric, by the degree of X. In a sequel to this paper our height conjecture is established for any projective diagonal Fano hypersurface, by exploiting a more general logarithmic setup.

math.AG↗

arXiv2024

Sharp bounds on the height of K-semistable Fano varieties II, the log case

In our previous work we conjectured - inspired by an algebro-geometric result of Fujita - that the height of an arithmetic Fano variety X of relative dimension $n$ is maximal when X is the projective space $\mathbb{P}^n_{\mathbb{Z}}$ over the integers, endowed with the Fubini-Study metric, if the corresponding complex Fano variety is K-semistable. In this work the conjecture is settled for diagonal hypersurfaces in $\mathbb{P}^{n+1}_{\mathbb{Z}}$. The proof is based on a logarithmic extension of our previous conjecture, of independent interest, which is established for toric log Fano varieties of relative dimension at most three, hyperplane arrangements on $\mathbb{P}^n_{\mathbb{Z}}$, as well as for general arithmetic orbifold Fano surfaces.

math.AG↗

arXiv2023

On K-stability, height bounds and the Manin-Peyre conjecture

This note discusses some intriguing connections between height bounds on complex K-semistable Fano varieties X and Peyre's conjectural formula for the density of rational points on X. Relations to an upper bound for the smallest rational point, proposed by Elsenhans-Jahnel, are also explored. These relations suggest an analog of the height inequalities, adapted to the real points, which is established for the real projective line and related to Kähler-Einstein metrics.

math.AG↗

arXiv2022

Kähler-Einstein metrics and Archimedean zeta functions

While the existence of a unique Kähler-Einstein metric on a canonically polarized manifold X was established by Aubin and Yau already in the 70s there are only a few explicit formulas available. In previous work a probabilistic construction of the Kähler-Einstein metric was introduced - involving canonical random point processes on X - which yields canonical approximations of the Kähler-Einstein metric, expressed as explicit period integrals over a large number of products of X. Here it is shown that the conjectural extension to the case when X is a Fano variety suggests a zero-free property of the Archimedean zeta functions defined by the partition functions of the probabilistic model. A weaker zero-free property is also shown to be relevant for the Calabi-Yau equation. The convergence in the case of log Fano curves is settled, exploiting relations to the complex Selberg integral in the orbifold case. Some intriguing relations to the zero-free property of the local automorphic L-functions appearing in the Langlands program and arithmetic geometry are also pointed out. These relations also suggest a natural p-adic extension of the probabilistic approach.

math.DG↗

arXiv2022

Measure preserving holomorphic vector fields, invariant anti-canonical divisors and Gibbs stability

Let X be a compact complex manifold whose anti-canonical line bundle is big. We show that X admits no non-trivial holomorphic vector fields if it is Gibbs stable (at any level). The proof is based on a vanishing result for measure preserving holomorphic vector fields on X of independent interest. As an application it shown that, in general, if the anti-canonical line bundle is big, there are no holomorphic vector fields on X that are tangent to a non-singular irreducible anti-canonical divisor S on X. More generally, the result holds for varieties with log terminal singularities and log pairs. Relations to a result of Berndtsson about generalized Hamiltonians and coercivity of the quantized Ding functional are also pointed out.

math.AG↗

arXiv2021

Priors leading to well-behaved Coulomb and Riesz gases versus zeroth-order phase transitions -- a potential-theoretic characterization

We give a potential-theoretic characterization of priors which have the property that the corresponding Coulomb gas is "well-behaved" and similarly for more general Riesz gases. This means that the laws of the empirical measures of the corresponding random point process satisfy a Large Deviation Principle with a rate functional which depends continuously on the temperature, in the sense of Gamma-convergence. Equivalently, there is no zeroth-order phase transition at zero temperature, in the mean field regime. This is shown to be the case for the Hausdorff measure on a compact Lipschitz hypersurface, as well as Lesbesgue measure on a bounded Lipschitz domain. We also provide constructions of priors, absolutely continuous with respect to Lebesgue measure on a smoothly bounded domain, such that the corresponding 2d Coulomb exhibits a zeroth-order phase transition. This is based on relations to Ullman's criterion in the theory of orthogonal polynomials and Bernstein-Markov inequalities.

math-ph↗

arXiv2021

The spherical ensemble and quasi-Monte-Carlo designs

The spherical ensemble is a well-known ensemble of N repulsive points on the two-dimensional sphere, which can realized in various ways (as a random matrix ensemble, a determinantal point process, a Coulomb gas, a Quantum Hall state...). Here we show that the spherical ensemble enjoys nearly optimal convergence properties from the point of view of numerical integration. More precisely, it is shown that the numerical integration rule corresponding to N nodes on the two-dimensional sphere sampled in the spherical ensemble is, with overwhelming probability, nearly a quasi-Monte-Carlo design in the sense of Brauchart-Saff-Sloan-Womersley (for any smoothness parameter s less than or equal to two). The key ingredient is a new explicit concentration of measure inequality for the spherical ensemble.

math.PR↗

arXiv2021

The probabilistic vs the quantization approach to Kähler-Einstein geometry

In the probabilistic construction of Kähler-Einstein metrics on a complex projective algebraic manifold X - involving random point processes on X - a key role is played by the partition function. In this work a new quantitative bound on the partition function is obtained. It yields, in particular, a new direct analytic proof that X admits a Kähler-Einstein metrics if it is uniformly Gibbs stable. The proof makes contact with the quantization approach to Kähler-Einstein geometry.

math.DG↗

arXiv2021

Emergent complex geometry

This is a double exposure of the probabilistic construction of Kahler-Einstein metrics on a complex projective algebraic variety X - where the Kahler-Einstein metric emerges from a canonical random point process on X - and the variational approach to the Yau-Tian-Donaldson conjecture, highlighting their connections. The final section is a report on joint work in progress with Sébastien Boucksom and Mattias Jonsson on how the non-Archimedean geometry of X (with respect to the trivial absolute value) also emerges from the probabilistic framework.

math.DG↗

arXiv2021

Ensemble equivalence for mean field models and plurisubharmonicity

We show that entropy is globally concave with respect to energy for a rich class of mean field interactions, including regularizations of the the point-vortex model in the plane, plasmas and self-gravitating matter in 2D, as well as the higher dimensional logarithmic interactions appearing in conformal geometry and power laws. The proofs are based on a corresponding "microscopic" concavity result at finite N, shown by leveraging an unexpected link to Kahler geometry and plurisubharmonic functions. Under more restrictive homogeneity assumptions strict concavity is obtained using a uniqueness result for free energy minimizers, established in a companion paper. The results imply that thermodynamic equivalence of ensembles holds for this class of mean field models. As an application it is shown that the critical inverse negative temperatures - in the macroscopic as well as the microscopic setting - coincide with the asymptotic slope of the corresponding microcanonical entropies. Along the way we also extend previous results on the thermodynamic equivalence of ensembles for continuous weakly positive definite interactions, concerning positive temperature states, to the general non-continuous case. In particular, singular situations are exhibited where, somewhat surprisingly, thermodynamic equivalence of ensembles fails at energy levels sufficiently close to the minimum energy level.

math-ph↗

arXiv2020

Convergence rates for discretized Monge-Ampère equations and quantitative stability of optimal transport

In recent works - both experimental and theoretical - it has been shown how to use computational geometry to efficently construct approximations to the optimal transport map between two given probability measures on Euclidean space, by discretizing one of the measures. Here we provide a quantative convergence analysis for the solutions of the corresponding discretized Monge-Ampère equations. This yields L^{2}-converge rates, in terms of the corresponding spatial resolution h, of the discrete approximations of the optimal transport map, when the source measure is discretized and the target measure has bounded convex support. Periodic variants of the results are also established. The proofs are based on quantitative stability results for optimal transport maps, shown using complex geometry.

math.NA↗

arXiv2020

Emergent Sasaki-Einstein geometry and AdS/CFT

We consider supergravity in five-dimensional Anti-De Sitter space $AdS_{5}$ with minimal supersymmetry, encoded by a Sasaki-Einstein metric on a five-dimensional compact manifold $M$. Our main result reveals how the Sasaki-Einstein metric emerges from a canonical state in the dual CFT, defined by a superconformal gauge theory in four dimensional Minkowski space $\mathbb{R}^{3,1}$in the t'Hooft limit where the rank $N$ tends to infinity. We obtain explicit finite $N-$approximations to the Sasaki-Einstein metric, expressed in terms of a canonical (i.e. background free) BPS-state on the gauge theory side. We also provide a string theory interpretation of the BPS-state in question, which sheds new light on the previously noted intriguing duality of giant gravitons.

hep-th↗

arXiv2020

The Sinkhorn algorithm, parabolic optimal transport and geometric Monge-Ampère equations

We show that the discrete Sinkhorn algorithm - as applied in the setting of Optimal Transport on a compact manifold - converges to the solution of a fully non-linear parabolic PDE of Monge-Ampere type, in a large-scale limit. The latter evolution equation has previously appeared in different contexts (e.g. on the torus it can be be identified with the Ricci flow). This leads to algorithmic approximations of the potential of the Optimal Transport map, as well as the Optimal Transport distance, with explicit bounds on the arithmetic complexity of the construction and the approximation errors. As applications we obtain explicit schemes of nearly linear complexity, at each iteration, for optimal transport on the torus and the two-sphere, as well as the far-field antenna problem. Connections to Quasi-Monte Carlo methods are exploited.

math.AP↗