arXiv subjects
Publications and source records attributed to Eleonora Di Nezza.
We study the geometric regularization of a positive closed current by the (twisted) K\"ahler-Ricci flow on a compact K\"ahler manifold. We conjecture that the local Arnold multiplicities linearly decrease to zero, while the flow produces complete K\"ahler metrics in the Zariski open subset of points that have small Lelong numbers. We prove this conjecture in complex dimension 1 and provide several partial results in higher dimension.
We study the geometric regularization of positive closed currents by the K\"ahler-Ricci flow on compact K\"ahler manifolds. In a previous work of ours, it was shown that the K\"ahler-Ricci flow immediately smoothes out such a current when it has zero Lelong numbers. We study here the case when $T_0$ has divisorial singularities, showing that the flow gradually replaces the latter by Poincar\'e type ones, providing an approximation of $T_0$ by complete K\"ahler metrics with bounded curvature in a Zariski open set.
We study continuity, H\"older regularity, and $C^{1,1}$-regularity of geodesics between continuous plurisubharmonic functions on bounded domains of $\mathbb{C}^n$. We then derive regularity properties of rooftop envelopes.
The space of K\"ahler potentials can be quantized through the classical Fubini-Study map, relating infinite-dimensional geometric structures to finite-dimensional symmetric spaces. We prove (exactly) when the Fubini-Study image of a geodesic line in the space of positive definite Hermitian matrices gives rise to a quasi-geodesic in the space of K\"ahler potentials. Furthermore, we introduce a quantization procedure for geodesics between potentials on normal K\"ahler varieties and show how this construction extends to the Sasaki setting.
These are notes of lectures given by the first named author during the CIME Summer school Calabi-Yau varieties. We survey known results concerning the complex Monge-Amp\`ere equations in Hermitian contexts obtained by many authors during the last fifteen years.
In this paper we investigate the existence of metrics with weighted constant scalar curvature (wcscK for short) on a compact K\"ahler manifold $X$: this notion include constant scalar curvature K\"ahler metrics, weighted solitons, Calabi's extremal K\"ahler metrics and extremal metric on semisimple principal fibrations. We prove that the coercivity of the weighted Mabuchi functional implies the existence of a wcscK metric, thereby achieving the equivalence. \\ We then give several applications in K\"ahler and toric geometry, such as a weighted version of the toric Yau-Tian-Donaldson correspondence, and the characterization of the existence of wcscK metric on total space of semisimple principal fibration $Y$ in term of existence of wcscK metric on its fiber $X$.
We study the Mabuchi functional associated to a big cohomology class. We define an invariant associated to transcendental Fujita approximations, whose vanishing is related to the Yau-Tian Donaldson conjecture. Assuming vanishing (finiteness) of this invariant we establish (almost) convexity along weak geodesics. As an application, we give an explicit expression of the distance $d_p$ in the big setting for finite entropy potentials.
Let $X$ be a compact K\"ahler manifold. In this paper we study the existence of constant weighted scalar curvature K\"ahler (weighted cscK) metrics on $X$. More precisely, we establish a priori $C^{k}$-estimates ($k\geq 0$) for the K\"ahler potential associated with these metrics, thereby extending a result due to Chen and Cheng in the classical cscK setting.
In this note, we generalize the notion of entropy for potentials in a relative full Monge-Ampère mass $\mathcal{E}(X, θ, ϕ)$, for a model potential $ϕ$. We then investigate stability properties of this condition with respect to blow-ups and perturbation of the cohomology class. We also prove a Moser-Trudinger type inequality with general weight and we show that functions with finite entropy lie in a relative energy class $\mathcal{E}^{\frac{n}{n-1}}(X, θ, ϕ)$ (provided $n>1$), while they have the same singularities of $ϕ$ when $n=1$.
Given a compact K\"ahler manifold, we survey the study of complex Monge-Amp\`ere 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-Amp\`ere operator.
Let $X$ be a compact complex manifold of complex dimension $n$ and $α$ be a smooth closed real form on $X$ such that its cohomology class $\{ α\}\in H^{1,1}(X, \mathbb{R})$ is big. In this paper we prove that, given a bounded function $f$ with bounded distributional laplacian in $X,$ the $α$-psh envelope $P(f)$ is also locally bounded with locally bounded distributional laplacian on the ample locus of $\{α\}.$
We prove a geodesic distance formula for quasi-psh functions with finite entropy, extending results by Chen and Darvas. We work with big and nef cohomology classes: a key result we establish is the convexity of the $K$-energy in this general setting. We then study Monge-Ampère measures on contact sets, generalizing a recent result by the first author and Trapani.
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.
Probability measures with either finite Monge-Ampère energy or finite entropy have played a central role in recent developments in Kähler geometry. In this note we make a systematic study of quasi-plurisubharmonic potentials whose Monge-Ampère measures have finite entropy. We show that these potentials belong to the finite energy class ${\mathcal E}^{\frac{n}{n-1}}$, where $n$ denotes the complex dimension, and provide examples showing that this critical exponent is sharp. Our proof relies on refined Moser-Trudinger inequalities for quasi-plurisubharmonic functions.
Refining Yau's and Kolodziej's techniques, we establish very precise uniform a priori estimates for degenerate complex Monge-Amp\`ere equations on compact K\"ahler manifolds, that allow us to control the blow up of the solutions as the cohomology class and the complex structure both vary. We apply these estimates to the study of various families of possibly singular K\"ahler varieties endowed with twisted K\"ahler-Einstein metrics, by analyzing the behavior of canonical densities, establishing uniform integrability properties, and developing the first steps of a pluripotential theory in families. This provides interesting information on the moduli space of stable varieties, extending works by Berman-Guenancia and Song, as well as on the behavior of singular Ricci flat metrics on (log) Calabi-Yau varieties, generalizing works by Rong-Ruan-Zhang, Gross-Tosatti-Zhang, Collins-Tosatti and Tosatti-Weinkove-Yang.
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.
Let $(X, ω)$ be a compact Kähler manifold of complex dimension n and $θ$ be a smooth closed real $(1,1)$-form on $X$ such that its cohomology class $\{ θ\}\in H^{1,1}(X, \mathbb{R})$ is pseudoeffective. Let $φ$ be a $θ$-psh function, and let $f$ be a continuous function on $X$ with bounded distributional laplacian with respect to $ω$ such that $φ\leq f. $ Then the non-pluripolar measure $θ_φ^n:= (θ+ dd^c φ)^n$ satisfies the equality: $$ {\bf{1}}_{\{ φ= f \}} \ θ_φ^n = {\bf{1}}_{\{ φ= f \}} \ θ_f^n,$$ where, for a subset $T\subseteq X$, ${\bf{1}}_T$ is the characteristic function. In particular we prove that \[ θ_{P_θ(f)}^n= { \bf {1}}_{\{P_θ(f) = f\}} \ θ_f^n\qquad {\rm and }\qquad θ_{P_θ[φ](f)}^n = { \bf {1}}_{\{P_θ[φ](f) = f \}} \ θ_f^n. \]
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.