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Yanir A. Rubinstein

Publications and source records attributed to Yanir A. Rubinstein.

At least 19 recordsLinked to original sources
arXiv2026

Quantitative Stability and Numerical Resolution of the Moment Measure Problem

The moment measure problem consists in finding a convex function $ψ$ whose moment measure, i.e., the pushforward by $\nabla ψ$ of the measure with density $e^{-ψ(\,\cdot\,)}$, is prescribed. It is highly non-linear and less understood than the related optimal transport problem. We establish a quantitative stability estimate for this problem. This estimate validates, as well as leads us to introduce, an approach to the numerical resolution of the moment measure problem inspired by semi-discrete optimal transport, consisting in approximating the prescribed measure by a finitely supported one. We describe a Newton method for solving the discrete problem thus obtained, and perform numerical experiments, studying the experimental rates of convergence of the approximation beyond the predictions of the stability estimate.

math.FA↗
arXiv2025

Symmetry notions for toric Fanos

We survey various notions of symmetry for toric varieties. These notions range from algebraic geometric, complex geometric, representation theoretic, combinatorial, convex geometric, to geometric stability. The main theorem gives the relationship between these notions. While mostly folklore knowledge, this does not seem to be readily available in the literature. Finally, we take the opportunity to give an accessible and simplified proof of Demazure's 1970 structure theorem for the automorphism group of a smooth toric variety, previously considered quite inaccessible.

math.AG↗
arXiv2025

Convexity and the degenerate special Lagrangian equation

In 2015 Rubinstein--Solomon introduced the degenerate special Lagrangian equation (DSL) that governs geodesics in the space of positive Lagrangians, showed that subsolutions in the top branch of DSL are convex in space, and raised the question of whether they should be convex in space-time and whether subsolutions in the second branch possess any convexity properties. In 2019, Darvas--Rubinstein gave a partial answer to the first problem by showing subsolutions in the top branch must be bi-convex. We settle both questions. The key new ingredient is a space-time coordinate transformation that preserves the space-time Lagrangian angle and allows for a partial $C^2$ estimate. This also shows that the top two branches of the DSL subequation have a $\star$-product structure in the sense of Ross--Witt-Nyström.

math.DG↗
arXiv2025

Stability thresholds for big classes

In 1987, the $α$-invariant theorem gave a fundamental criterion for existence of Kahler-Einstein metrics on smooth Fano manifolds. In 2012, Odaka-Sano extended the framework to $\mathbb{Q}$-Fano varieties in terms of K-stability, and in 2017 Fujita related this circle of ideas to the $δ$-invariant of Fujita-Odaka. We introduce new invariants on the big cone and prove a generalization of the Tian-Odaka-Sano Theorem to all big classes on varieties with klt singularities, and moreover for all volume quantiles $τ\in[0,1]$. The special degenerate (collapsing) case $τ=0$ on ample classes recovers Odaka-Sano's theorem. This leads to many new twisted Kahler-Einstein metrics on big classes. Of independent interest, the proof involves a generalization to sub-barycenters of the classical Neumann-Hammer Theorem from convex geometry.

math.DG↗
arXiv2024

Asymptotics of quantized barycenters of lattice polytopes with applications to algebraic geometry

This article addresses a combinatorial problem with applications to algebraic geometry. To a convex lattice polytope $P$ and each of its integer dilations $kP$ one may associate the barycenter of its lattice points. This sequence of $k$-quantized barycenters converge to the (classical) barycenter of the polytope considered as a convex body. A basic question arises: is there a complete asymptotic expansion for this sequence? If so, what are its terms? This article initiates the study of this question. First, we establish the existence of such an expansion as well as determine the first two terms. Second, for Delzant lattice polytopes we use toric algebra to determine all terms using mixed volumes of virtual rooftop polytopes, or alternatively in terms of higher Donaldson--Futaki invariants. Third, for reflexive polytopes we show the quantized barycenters are colinear to first order, and actually colinear in the case of polygons. The proofs use Ehrhart theory, convexity arguments, and toric algebra. As applications we derive the complete asymptotic expansion of the Fujita--Odaka stability thresholds $δ_k$ on arbitrary polarizations on (possibly singular) toric varieties. In fact, we show they are rational functions of $k$ for sufficiently large $k$. This gives the first general result on Tian's stabilization problem for $δ_k$-invariants for (possibly singular) toric Fanos: $δ_k$ stabilize in $k$ if and only if they are all equal to $1$, and when smooth if and only if asymptotically Chow semistable. We also relate the asymptotic expansions to higher Donaldson--Futaki invariants of test configurations motivated by Ehrhart theory, and unify in passing previous results of Donaldson, Ono, Futaki, and Rubinstein--Tian--Zhang.

math.AG↗
arXiv2024

Convex meets complex

Convex geometry and complex geometry have long had fascinating interactions. This survey offers a tour of a few.

math.CV↗
arXiv2024

Two-dimensional Błocki, $L^p$-Mahler, and Bourgain conjectures

We confirm, in dimension two, Blocki's conjectures on sharp lower bounds for Bergman kernels of tube domains. To that end, we verify a broader class of $L^p$-Mahler conjectures due to Berndtsson and the authors, where $p=1$ are Blocki's conjecture, and $p=\infty$ are Mahler's conjectures. The proofs are technically challenging as the $L^p$-Mahler volume is considerably harder to deal with analytically compared to Mahler's volume, and furthermore duality is lost. In addition, unlike in the classical Mahler setting, the non-symmetric setting is considerably more involved than the symmetric one. The proofs involve studying the effect of Mahler's classical sliding of vertices on two-dimensional polytopes on the $L^p$-polar body (no longer a polytope). Some arguments are inspired by works of Campi--Gronchi and Meyer--Reisner on volumes of classical polar bodies of shadow systems. In passing, we also explore how Mahler's sliding affects the isotropic constant. This leads to an elementary proof of Bourgain's strong hyperplane conjectures in dimension two, originally due to Bisztriczky--Böröczky, Campi--Colesanti--Gronchi and Meckes. Specifically, we show that, as a function of the sliding parameter, the isotropic constant raised to an appropriate power is a convex quadratic polynomial.

math.FA↗
arXiv2024

Tian's stabilization problem for toric Fanos

In 1988, Tian posed the stabilization problem for equivariant global log canonical thresholds. We solve it in the case of toric Fano manifolds. This is the first general result on Tian's problem. A key new estimate involves expressing complex singularity exponents associated to orbits of a group action in terms of support and gauge functions from convex geometry. These techniques also yield a resolution of another conjecture of Tian from 2012 on more general thresholds associated to Grassmannians of plurianticanonical series.

math.AG↗
arXiv2024

Eguchi--Hanson metrics arising from Kahler--Einstein edge metrics

Calabi--Hirzebruch manifolds are higher-dimensional generalizations of both the football and Hirzebruch surfaces. We construct a family of Kahler--Einstein edge metrics singular along two disjoint divisors on the Calabi--Hirzebruch manifolds and study their Gromov--Hausdorff limits when either cone angle tends to its extreme value. As a very special case, we show that the celebrated Eguchi--Hanson metric arises in this way naturally as a Gromov--Hausdorff limit. We also completely describe all other (possibly rescaled) Gromov--Hausdorff limits which exhibit a wide range of behaviors, resolving in this setting a conjecture of Cheltsov--Rubinstein. This gives a new interpretation of both the Eguchi--Hanson space and Calabi's Ricci flat spaces as limits of compact singular Einstein spaces.

math.DG↗
arXiv2023

$L^p$-polarity, Mahler volumes, and the isotropic constant

This article introduces $L^p$ versions of the support function of a convex body $K$ and associates to these canonical $L^p$-polar bodies $K^{\circ, p}$ and Mahler volumes $\mathcal{M}_p(K)$. Classical polarity is then seen as $L^\infty$-polarity. This one-parameter generalization of polarity leads to a generalization of the Mahler conjectures, with a subtle advantage over the original conjecture: conjectural uniqueness of extremizers for each $p\in(0,\infty)$. We settle the upper bound by demonstrating the existence and uniqueness of an $L^p$-Santaló point and an $L^p$-Santaló inequality for symmetric convex bodies. The proof uses Ball's Brunn--Minkowski inequality for harmonic means, the classical Brunn--Minkowski inequality, symmetrization, and a systematic study of the $\mathcal{M}_p$ functionals. Using our results on the $L^p$-Santaló point and a new observation motivated by complex geometry, we show how Bourgain's slicing conjecture can be reduced to lower bounds on the $L^p$-Mahler volume coupled with a certain conjectural convexity property of the logarithm of the Monge--Ampère measure of the $L^p$-support function. We derive a suboptimal version of this convexity using Kobayashi's theorem on the Ricci curvature of Bergman metrics to illustrate this approach to slicing. Finally, we explain how Nazarov's complex analytic approach to the classical Mahler conjecture is instead precisely an approach to the $L^1$-Mahler conjecture.

math.FA↗
arXiv2022

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

The Chebyshev potential of a Kähler potential on 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 curve in the space of Kähler potentials is linear in the time variable if and only if the latter curve 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 goal of this article is to disprove this conjecture. More generally, we characterize the Fubini--Study geodesics for which the conjecture is true on 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.

math.CV↗
arXiv2022

Classification of strongly asymptotically log del Pezzo flags and surfaces

We introduce the notion of strongly asymptotically log del Pezzo flags, and classify such flags under the assumption that their zero-dimensional part lies in the boundary. We use this result to give a new and conceptual proof of the classification of strongly asymptotically log del Pezzo surfaces, originally due to Cheltsov and the author.

math.AG↗
arXiv2022

The Nazarov proof of the non-symmetric Bourgain--Milman inequality

In 2012, Nazarov used Bergman kernels and Hormander's $L^2$ estimates for the $\bar\partial$-equation to give a new proof of the Bourgain--Milman theorem for symmetric convex bodies and made some suggestions on how his proof should extend to general convex bodies. This article achieves this extension and serves simultaneously as an exposition to Nazarov's work. A key new ingredient is an affine invariant associated to the Bergman kernel of a tube domain. This gives the first `complex' proof of the Bourgain--Milman theorem for general convex bodies, specifically, without using symmetrization.

math.FA↗
arXiv2022

On large deviation principles and the Monge--Ampère equation (following Berman, Hultgren)

This is mostly an exposition, aimed to be accessible to geometers, analysts, and probabilists, of a fundamental recent theorem of R. Berman with recent developments by J. Hultgren, that asserts that the second boundary value problem for the real Monge--Ampère equation admits a probabilistic interpretation, in terms of many particle limit of permanental point processes satisfying a large deviation principle with a rate function given explicitly using optimal transport. An alternative proof of a step in the Berman--Hultgren Theorem is presented allowing to to deal with all "tempratures" simultaneously instead of first reducing to the zero-temperature case.

math.PR↗
arXiv2022

On the body of ample angles of asymptotically log Fano varieties

In dimension two, we reduce the classification problem for asymptotically log Fano pairs to the problem of determining generality conditions on certain blow-ups. In any dimension, we prove the rationality of the body of ample angles of an asymptotically log Fano pair, i.e., these convex bodies are always rational polytopes.

math.AG↗
arXiv2021

Angle deformation of Kähler-Einstein edge metrics on Hirzebruch surfaces

We construct a family of Kähler-Einstein edge metrics on all Hirzebruch surfaces using the Calabi ansatz and study their angle deformation. This allows us to verify in some special cases a conjecture of Cheltsov-Rubinstein that predicts convergence towards a non-compact Calabi-Yau fibration in the small angle limit. We also give an example of a Kähler-Einstein edge metric whose edge singularity is rigid, answering a question posed by Cheltsov.

math.DG↗
arXiv2021

Basis divisors and balanced metrics

Using log canonical thresholds and basis divisors Fujita--Odaka introduced purely algebro-geometric invariants $δ_m$ whose limit in $m$ is now known to characterize uniform K-stability on a Fano variety. As shown by Blum-Jonsson this carries over to a general polarization, and together with work of Berman, Boucksom, and Jonsson, it is now known that the limit of these $δ_m$-invariants characterizes uniform Ding stability. A basic question since Fujita-Odaka's work has been to find an analytic interpretation of these invariants. We show that each $δ_m$ is the coercivity threshold of a quantized Ding functional on the $m$-th Bergman space and thus characterizes the existence of balanced metrics. This approach has a number of applications. The most basic one is that it provides an alternative way to compute these invariants, which is new even for $\mathbb{P}^n$. Second, it allows us to introduce algebraically defined invariants that characterize the existence of Kähler-Ricci solitons (and the more general $g$-solitons of Berman-Witt Nyström), as well as coupled versions thereof. Third, it leads to approximation results involving balanced metrics in the presence of automorphisms that extend some results of Donaldson.

math.DG↗