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Tamás Darvas

Publications and source records attributed to Tamás Darvas.

At least 37 records · Page 2Linked to original sources
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↗
arXiv2020

Log-concavity of volume and complex Monge-Ampère equations with prescribed singularity

Let $(X,ω)$ be a compact Kähler manifold. We prove the existence and uniqueness of solutions to complex Monge-Ampère equations with prescribed singularity type. Compared to previous work, the assumption of small unbounded locus is dropped, and we work with general model type singularities. We state and prove our theorems in the context of big cohomology classes, however our results are new in the Kähler case as well. As an application we confirm a conjecture by Boucksom-Eyssidieux-Guedj-Zeriahi concerning log-concavity of the volume of closed positive $(1,1)$-currents. Finally, we show that log-concavity of the volume in complex geometry corresponds to the Brunn-Minkowski inequality in convex geometry, pointing out a dictionary between our relative pluripotential theory and $P$-relative convex geometry. Applications related to stability and existence of csck metrics are treated elsewhere.

math.DG↗
arXiv2019

The metric geometry of singularity types

Let $X$ be a compact Kähler manifold. Given a big cohomology class $\{θ\}$, there is a natural equivalence relation on the space of $θ$-psh functions giving rise to $\mathcal S(X,θ)$, the space of singularity types of potentials. We introduce a natural pseudometric $d_{\mathcal {S}}$ on $\mathcal S(X,θ)$ that is non-degenerate on the space of model singularity types and whose atoms are exactly the relative full mass classes. In the presence of positive mass we show that this metric space is complete. As applications, we show that solutions to a family of complex Monge-Ampère equations with varying singularity type converge as governed by the $d_\mathcal S$-topology, and we obtain a semicontinuity result for multiplier ideal sheaves associated to singularity types, extending the scope of previous results from the local context.

math.DG↗
arXiv2019

The isometries of the space of Kähler metrics

Given a compact Kähler manifold, we prove that all global isometries of the space of Kähler metrics are induced by biholomorphisms and anti-biholomorphisms of the manifold. In particular, there exist no global symmetries for Mabuchi's metric. Moreover, we show that the Mabuchi completion does not even admit local symmetries. Closely related to these findings, we provide a large class of metric geodesic segments that can not be extended at one end, pointing out the first such examples in the literature.

math.DG↗
arXiv2018

Regularity of weak minimizers of the K-energy and applications to properness and K-stability

Let $(X,ω)$ be a compact Kähler manifold and $\mathcal H$ the space of Kähler metrics cohomologous to $ω$. If a cscK metric exists in $\mathcal H$, we show that all finite energy minimizers of the extended K-energy are smooth cscK metrics, partially confirming a conjecture of Y.A. Rubinstein and the second author. As an immediate application, we obtain that existence of a cscK metric in $\mathcal H$ implies J-properness of the K-energy, thus confirming one direction of a conjecture of Tian. Exploiting this properness result we prove that an ample line bundle $(X,L)$ admitting a cscK metric in $c_1(L)$ is $K$-polystable.

math.DG↗
arXiv2018

Quantization in geometric pluripotential theory

The space of Kähler metrics can, on the one hand, be approximated by subspaces of algebraic metrics, while, on the other hand, can be enlarged to finite-energy spaces arising in pluripotential theory. The latter spaces are realized as metric completions of Finsler structures on the space of Kähler metrics. The former spaces are the finite-dimensional spaces of Fubini--Study metrics of Kähler quantization. The goal of this article is to draw a connection between the two. We show that the Finsler structures on the space of Kähler potentials can be quantized. More precisely, given a Kähler manifold polarized by an ample line bundle we endow the space of Hermitian metrics on powers of that line bundle with Finsler structures and show that the resulting path length metric spaces recover the corresponding metric completions of the Finsler structures on the space of Kähler potentials. This has a number of applications, among them a new approach to the rooftop envelopes and Pythagorean formulas of Kähler geometry, a new Lidskii type inequality on the space of Kähler metrics, and approximation of finite energy potentials, as well as geodesic segments by the corresponding smooth algebraic objects.

math.DG↗
arXiv2018

Monotonicity of non-pluripolar products and complex Monge-Ampère equations with prescribed singularity

We establish the monotonicity property for the mass of non-pluripolar products on compact Kahler manifolds, and we initiate the study of complex Monge-Ampere type equations with prescribed singularity type. Using the variational method of Berman-Boucksom-Guedj-Zeriahi we prove existence and uniqueness of solutions with small unbounded locus. We give applications to Kahler-Einstein metrics with prescribed singularity, and we show that the log-concavity property holds for non-pluripolar products with small unbounded locus.

math.DG↗
arXiv2017

Convergence of the Kähler-Ricci iteration

The Ricci iteration is a discrete analogue of the Ricci flow. According to Perelman, the Ricci flow converges to a Kahler-Einstein metric whenever one exists, and it has been conjectured that the Ricci iteration should behave similarly. This article confirms this conjecture. As a special case, this gives a new method of uniformization of the Riemann sphere.

math.DG↗
arXiv2017

Weak geodesic rays in the space of Kähler potentials and the class E(X,ω)

Suppose (X,ω) is a compact Kähler manifold. In the present work we propose a simple construction for weak geodesic rays in the space of Kähler metrics that seems to be tied together with properties of the class E(X,ω). As an application of our construction, we prove a characterization of E(X,ω) in terms of envelopes.

math.CV↗
arXiv2017

On the singularity type of full mass currents in big cohomology classes

Let $X$ be a compact Kähler manifold and $\{θ\}$ be a big cohomology class. We prove several results about the singularity type of full mass currents, answering a number of open questions in the field. First, we show that the Lelong numbers and multiplier ideal sheaves of $θ$-plurisubharmonic functions with full mass are the same as those of the current with minimal singularities. Second, given another big and nef class $\{η\}$, we show the inclusion $\mathcal{E}(X,η) \cap {PSH}(X,θ) \subset \mathcal{E}(X,θ).$ Third, we characterize big classes whose full mass currents are "additive". Our techniques make use of a characterization of full mass currents in terms of the envelope of their singularity type. As an essential ingredient we also develop the theory of weak geodesics in big cohomology classes. Numerous applications of our results to complex geometry are also given.

math.DG↗
arXiv2016

Metric geometry of normal Kähler spaces, energy properness, and existence of canonical metrics

Let $(X,ω)$ be a compact normal Kähler space, with Hodge metric $ω$. In this paper, the last in a sequence of works studying the relationship between energy properness and canonical Kähler metrics, we introduce a geodesic metric structure on $\mathcal H_ω(X)$, the space of Kähler potentials, whose completion is the finite energy space $\mathcal E^1_ω(X)$. Using this metric structure and the results of Berman-Boucksom-Eyssidieux-Guedj-Zeriahi as ingredients in the existence/properness principle of Rubinstein and the author, we show that existence of Kähler-Einstein metrics on log Fano pairs is equivalent to properness of the K-energy in a suitable sense. To our knowledge, this result represents the first characterization of general log Fano pairs admitting Kähler-Einstein metrics. We also discuss the analogous result for Kähler-Ricci solitons on Fano varieties.

math.DG↗
arXiv2016

Comparison of the Calabi and Mabuchi geometries and applications to geometric flows

Suppose $(X,ω)$ is a compact Kähler manifold. We introduce and explore the metric geometry of the $L^{p,q}$-Calabi Finsler structure on the space of Kähler metrics $\mathcal H$. After noticing that the $L^{p,q}$-Calabi and $L^{p'}$-Mabuchi path length topologies on $\mathcal H$ do not typically dominate each other, we focus on the finite entropy space $\mathcal E^{Ent}$, contained in the intersection of the $L^p$-Calabi and $L^1$-Mabuchi completions of $\mathcal H$ and find that after a natural strengthening, the $L^p$-Calabi and $L^1$-Mabuchi topologies coincide on $\mathcal E^{Ent}$. As applications to our results, we give new convergence results for the Kähler--Ricci flow and the weak Calabi flow.

math.DG↗
arXiv2016

A minimum principle for Lagrangian graphs

The classical minimum principle is foundational in convex and complex analysis and plays an important role in the study of the real and complex Monge-Ampere equations. This note establishes a minimum principle in Lagrangian geometry. This principle relates the classical Lagrangian angle of Harvey-Lawson and the space-time Lagrangian angle introduced recently by Rubinstein-Solomon. As an application, this gives a new formula for solutions of the degenerate special Lagrangian equation in space-time in terms of the (time) partial Legendre transform of a family of solutions of obstacle problems for the (space) non-degenerate special Lagrangian equation.

math.SG↗
arXiv2016

Convexity of the extended K-energy and the large time behaviour of the weak Calabi flow

Let $(X,ω)$ be a compact connected Kähler manifold and denote by $(\mathcal E^p,d_p)$ the metric completion of the space of Kähler potentials $\mathcal H_ω$ with respect to the $L^p$-type path length metric $d_p$. First, we show that the natural analytic extension of the (twisted) Mabuchi K-energy to $\mathcal E^p$ is a $d_p$-lsc functional that is convex along finite energy geodesics. Second, following the program of J. Streets, we use this to study the asymptotics of the weak (twisted) Calabi flow inside the CAT(0) metric space $(\mathcal E^2,d_2)$. This flow exists for all times and coincides with the usual smooth (twisted) Calabi flow whenever the latter exists. We show that the weak (twisted) Calabi flow either diverges with respect to the $d_2$-metric or it $d_1$-converges to some minimizer of the K-energy inside $\mathcal E^2$. This gives the first concrete result about the long time convergence of this flow on general Kähler manifolds, partially confirming a conjecture of Donaldson. Finally, we investigate the possibility of constructing destabilizing geodesic rays asymptotic to diverging weak (twisted) Calabi trajectories, and give a result in the case when the twisting form is Kähler. If the twisting form is only smooth, we reduce this problem to a conjecture on the regularity of minimizers of the K-energy on $\mathcal E^1$, known to hold in case of Fano manifolds.

math.DG↗
arXiv2015

Tian's properness conjectures and Finsler geometry of the space of Kahler metrics

Well-known conjectures of Tian predict that existence of canonical Kahler metrics should be equivalent to various notions of properness of Mabuchi's K-energy functional. In some instances this has been verified, especially under restrictive assumptions on the automorphism group. We provide counterexamples to the original conjecture in the presence of continuous automorphisms. The construction hinges upon an alternative approach to properness that uses in an essential way the metric completion with respect to a Finsler metric and its quotients with respect to group actions. This approach also allows us to formulate and prove new optimal versions of Tian's conjecture in the setting of smooth and singular Kahler-Einstein metrics, with or without automorphisms, as well as for Kahler-Ricci solitons. Moreover, we reduce both Tian's original conjecture (in the absence of automorphisms) and our modification of it (in the presence of automorphisms) in the general case of constant scalar curvature metrics to a conjecture on regularity of minimizers of the K-energy in the Finsler metric completion. Finally, our results also resolve Tian's conjecture on the Moser-Trudinger inequality for Fano manifolds with Kahler-Einstein metrics.

math.DG↗