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Tamás Darvas

Publications and source records attributed to Tamás Darvas.

At least 19 recordsLinked to original sources
arXiv2026

Demailly-Kollár continuity on klt pairs, and applications to alpha and delta invariants

We establish a Demailly-Kollár type continuity theorem for plurisubharmonic functions with respect to adapted measures on normal complex analytic klt pairs. As applications, we prove the equality of the analytic and divisorial versions of the alpha and delta invariants on compact normal Kähler klt pairs, thereby completing a program initiated by the second author. We further show that the two local alpha invariants introduced by Guedj and Trusiani for an isolated log terminal singularity coincide and that their common value is the $n$th root of Li's normalized volume. These identities have geometric consequences. The equality for the delta invariant yields a Yau-Tian-Donaldson type divisorial criterion for the solvability of twisted Kähler-Einstein equations in big cohomology classes, without a semipositivity assumption on the twist. The local alpha identity determines algebraically the critical exponent governing the existence of positively curved KE metrics near an isolated log terminal singularity, thus confirming a prediction of Guedj and Trusiani.

math.DG↗
arXiv2026

Lines in the space of Kähler metrics

We establish a Ross-Witt Nyström correspondence for weak geodesic lines in the (completed) space of Kähler metrics. We construct a wide range of weak geodesic lines on arbitrary projective Kähler manifolds that are not generated by holomorphic vector fields, in the process disproving a folklore conjecture popularized by Berndtsson. Remarkably, some of these weak geodesic lines turn out to be smooth. In the case of Riemann surfaces, our results can be significantly sharpened. Finally, we investigate the validity of Euclid's fifth postulate for the space of Kähler metrics.

math.DG↗
arXiv2026

The trace operator of quasi-plurisubharmonic functions on compact Kähler manifolds

We introduce the trace operator for quasi-plurisubharmonic functions on compact Kähler manifolds, allowing to study the singularities of such functions along submanifolds where their generic Lelong numbers vanish. Using this construction we obtain novel $L^2$ extension theorems and give applications to restricted volumes of big line bundles.

math.DG↗
arXiv2026

Sharp $C^{1,\bar1}$ estimates in Kähler quantization and non-pluripolar Radon measures

Let $K_φ$ denote the weighted Bergman kernel associated to a plurisubharmonic function $φ$. We obtain upper bounds and positive lower bounds for the Bergman metric $i\partial \bar{\partial} \log K_φ$, expressed solely in terms of upper bounds and positive lower bounds of $i\partial \bar{\partial}φ$. Our approach applies in both local and compact Kähler settings. As an immediate application we obtain the optimal $C^{1,α}$-convergence for the quantization of Kähler currents with bounded coefficients. We also show that any non-pluripolar Radon measure on a compact Kähler manifold admits a quantization.

math.DG↗
arXiv2026

Transcendental Okounkov bodies

We show that the volume of transcendental big $(1,1)$-classes on compact Kähler manifolds can be realized by convex bodies, thus answering questions of Lazarsfeld-Mustaţă and Deng. In our approach we use an approximation process by partial Okounkov bodies together with properties of the restricted volume, and we study the extension of Kähler currents, as well as the bimeromorphic behavior of currents with analytic singularities. We also establish a connection between transcendental Okounkov bodies and toric degenerations.

math.DG↗
arXiv2025

A YTD correspondence for constant scalar curvature metrics

Given a compact Kähler manifold, to better understand Mabuchi's $K$ energy we introduce a family of $K^β$ energies, whose favorable properties are similar to those of the Ding energy from the Fano case. The construction uses Berman's transcendental quantization, and we show that the slope of the $K^β$ energies along test configurations can be computed using intersection theory. With these ingredients in place we provide a uniform Yau-Tian-Donaldson correspondence that characterizes the existence of a unique constant scalar curvature Kähler metric using test configurations. Combining our techniques with the non-Archimedean approach to $K$-stability pioneered by Boucksom--Jonsson, we show that the properness of the classical $K$ energy can be tested by checking its slope along a distinguished subclass of Chi Li-type models, called log discrepancy models, thus yielding another $G$-uniform Yau--Tian--Donaldson correspondence.

math.DG↗
arXiv2024

The Hausdorff distance and metrics on toric singularity types

Given a compact Kähler manifold $(X,ω)$, due to the work of Darvas-Di Nezza-Lu, the space of singularity types of $ω$-psh functions admits a natural pseudo-metric $d_\mathcal S$ that is complete in the presence of positive mass. When restricted to model singularity types, this pseudo-metric is a bona fide metric. In case of the projective space, there is a known one-to-one correspondence between toric model singularity types and convex bodies inside the unit simplex. Hence in this case it is natural to compare the $d_\mathcal S$ metric to the classical Hausdorff metric. We provide precise Hölder bounds, showing that their induced topologies are the same. More generally, we introduce a quasi-metric $d_G$ on the space of compact convex sets inside an arbitrary convex body $G$, with $d_\mathcal S = d_G$ in case $G$ is the unit simplex. We prove optimal Hölder bounds comparing $d_G$ with the Hausdorff metric. Our analysis shows that the Hölder exponents differ depending on the geometry of $G$, with the worst exponents in case $G$ is a polytope, and the best in case $G$ has $C^2$ boundary.

math.CV↗
arXiv2024

A transcendental approach to non-Archimedean metrics of pseudoeffective classes

We introduce the concept of non-Archimedean metrics attached to a transcendental pseudoeffective cohomology class on a compact Kähler manifold. This is obtained via extending the Ross-Witt Nyström correspondence to the relative case, and we point out that our construction agrees with that of Boucksom-Jonsson when the class is induced by a pseudoeffective $\mathbb Q$-line bundle. We introduce the notion of a flag configuration attached to a transcendental big class, recovering the notion of a test configuration in the ample case. We show that non-Archimedean finite energy metrics are approximable by flag configurations, and very general versions of the radial Ding energy are continuous, a novel result even in the ample case. As applications, we characterize the delta invariant as the Ding semistability threshold of flag configurations and filtrations and prove a YTD type existence theorem in terms of flag configurations.

math.AG↗
arXiv2024

The volume of pseudoeffective line bundles and partial equilibrium

Let $(L,he^{-u})$ be a pseudoeffective line bundle on an $n$-dimensional compact Kähler manifold $X$. Let $h^0(X,L^k\otimes \mathcal I(ku))$ be the dimension of the space of sections $s$ of $L^k$ such that $h^k(s,s)e^{-ku}$ is integrable. We show that the limit of $k^{-n}h^0(X,L^k\otimes \mathcal I(ku))$ exists, and equals the non-pluripolar volume of $P[u]_\mathcal I$, the $\mathcal I$-model potential associated to $u$. We give applications of this result to Kähler quantization: fixing a Bernstein-Markov measure $ν$, we show that the partial Bergman measures of $u$ converge weakly to the non-pluripolar Monge--Ampère measure of $P[u]_\mathcal I$, the partial equilibrium.

math.DG↗
arXiv2024

Twisted Kähler-Einstein metrics in big classes

We prove existence of twisted Kähler-Einstein metrics in big cohomology classes, using a divisorial stability condition. In particular, when $-K_X$ is big, we obtain a uniform Yau-Tian-Donaldson existence theorem for Kähler-Einstein metrics. To achieve this, we build up from scratch the theory of Fujita-Odaka type delta invariants in the transcendental big setting, using pluripotential theory. We do not use the K-energy in our arguments, and our techniques provide a simple roadmap to prove Yau-Tian-Donaldson existence theorems for Kähler-Einstein type metrics, that only needs convexity of the appropriate Ding energy. As an application, we give a simplified proof of Li-Tian-Wang's existence theorem in the log Fano setting.

math.DG↗
arXiv2023

Relative pluripotential theory on compact Kähler manifolds

Given a compact Kähler manifold, we survey the study of complex Monge-Ampère type equations with prescribed singularity type, developed by the authors in a series of papers. In addition, we give a general answer to a question of Guedj--Zeriahi about the finite energy range of the complex Monge-Ampère operator.

math.CV↗
arXiv2023

The Mabuchi geometry of low energy classes

Let $(X,ω)$ be a Kähler manifold and $ψ: \Bbb R \to \Bbb R_+$ be a concave weight. We show that $\mathcal H_ω$ admits a natural metric $d_ψ$ whose completion is the low energy space $\mathcal E_ψ$, introduced by Guedj-Zeriahi. As $d_ψ$ is not induced by a Finsler metric, the main difficulty is to show that the triangle inequality holds. We study properties of the resulting complete metric space $(\mathcal E_ψ,d_ψ)$.

math.DG↗
arXiv2023

Extremizers of the $J$ functional with respect to the $d_1$ metric

In previous work, Darvas-George-Smith obtained inequalities between the large scale asymptotic of the $J$ functional with respect to the $d_1$ metric on the space of toric Kähler metrics/rays. In this work we prove sharpness of these inequalities on all toric Kähler manifolds, and study the extremizing potentials/rays. On general Kähler manifolds we show that existence of radial extremizers is equivalent with the existence of plurisupported currents, as introduced and studied by McCleerey.

math.DG↗
arXiv2022

Griffiths extremality, interpolation of norms, and Kähler quantization

Following Kobayashi, we consider Griffiths negative complex Finsler bundles, naturally leading us to introduce Griffiths extremal Finsler metrics. As we point out, this notion is closely related to the theory of interpolation of norms, and is characterized by an equation of complex Monge--Ampère type, whose corresponding Dirichlet problem we solve. As applications, we prove that Griffiths extremal Finsler metrics quantize solutions to a natural PDE in Kähler geometry, related to the construction of flat maps for the Mabuchi metric.

math.DG↗
arXiv2021

The closures of test configurations and algebraic singularity types

Given a Kähler manifold $X$ with an ample line bundle $L$, we consider the metric space of $L^1$ geodesic rays associated to the first Chern class $c_1(L)$. We characterize rays that can be approximated by ample test configurations. At the same time, we also characterize the closure of algebraic singularity types among all singularity types of quasi-plurisubharmonic functions, pointing out the very close relationship between these two seemingly unrelated problems. By Bonavero's holomorphic Morse inequalities, the arithmetic and non-pluripolar volumes of algebraic singularity types coincide. We show that in general the arithmetic volume dominates the non-pluripolar one, and equality holds exactly on the closure of algebraic singularity types. Analogously, we give an estimate for the Monge--Ampère energy of a general $L^1$ ray in terms of the arithmetic volumes along its Legendre transform. Equality holds exactly for rays approximable by test configurations. Various other cohomological and potential theoretic characterizations are given in both settings. As applications, we give a concrete formula for the non-Archimedean Monge--Ampère energy in terms of asymptotic expansion, and show the continuity of the projection map from $L^1$ rays to non-Archimedean rays. We also show that the closure of ample test configurations and filtrations gives the same set of rays.

math.DG↗
arXiv2021

L^1 metric geometry of big cohomology classes

Suppose $(X,ω)$ is a compact Kähler manifold of dimension $n$, and $θ$ is closed $(1,1)$-form representing a big cohomology class. We introduce a metric $d_1$ on the finite energy space $\mathcal{E}^1(X,θ)$, making it a complete geodesic metric space. This construction is potentially more rigid compared to its analog from the Kähler case, as it only relies on pluripotential theory, with no reference to infinite dimensional $L^1$ Finsler geometry. Lastly, by adapting the results of Ross and Witt Nyström to the big case, we show that one can construct geodesic rays in this space in a flexible manner.

math.DG↗
arXiv2021

Geometric pluripotential theory on Kähler manifolds

Finite energy pluripotential theory accommodates the variational theory of equations of complex Monge-Ampère type arising in Kähler geometry. Recently it has been discovered that many of the potential spaces involved have a rich metric geometry, effectively turning the variational problems in question into problems of infinite dimensional convex optimization, yielding existence results for solutions of the underlying complex Monge-Ampère equations. The purpose of this survey is to describe these developments from basic principles.

math.DG↗