arXiv subjects

Rolf Andreasson

Publications and source records attributed to Rolf Andreasson.

7 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

Critical temperatures and collapsing of two-dimensional Log gases

We consider the canonical ensemble of a system of point particles on the sphere interacting via a logarithmic pair potential. In this setting, we study the associated Gibbs measure and partition function, and we derive explicit formulas relating the critical temperature, at which the partition function diverges, to a certain discrete optimization problem. We further show that the asymptotic behavior of both the partition function and the Gibbs measure near the critical temperature is governed by the same optimization problem. Our approach relies on the Fulton--MacPherson compactification of configuration spaces and analytic continuation of complex powers. To illustrate the results, we apply them to well-studied systems, including the two-component plasma and the Onsager model of turbulence. In particular, for the two-component plasma with general charges, we describe the formation of dipoles close to the critical temperature, which we determine explicitly.

math-ph↗

arXiv2025

Regularity of the solution to a real Monge--Ampère equation on the boundary of a simplex

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.

math.AP↗

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

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

Solvability of Monge-Ampère equations and tropical affine structures on reflexive polytopes

Given a reflexive polytope with a height function, we prove a necessary and sufficient condition for solvability of the associated Monge-Ampère equation. When the polytope is Delzant, solvability of this equation implies the metric SYZ conjecture for the corresponding family of Calabi-Yau hypersurfaces. We show how the location of the singularities in the tropical affine structure is determined by the PDE in the spirit of a free boundary problem and give positive and negative examples, demonstrating subtle issues with both solvability and properties of the singular set. We also improve on existing results regarding the SYZ conjecture for the Fermat family by showing regularity of the limiting potential.

math.DG↗