arXiv subjects

Antonio Trusiani

Publications and source records attributed to Antonio Trusiani.

14 recordsLinked to original sources

arXiv2026

A solution to the Yau-Tian-Donaldson Conjecture through Special Fujita Approximations

We show that any big line bundle on a smooth projective variety admits a special Fujita approximation: the volume and the first Riemann-Roch coefficient are both approximated by those of ample $\mathbb{Q}$-line bundles on higher models. Exploiting previous works by Boucksom, Jonsson and Li, we solve the Boucksom-Jonsson Regularization Conjecture on the Non-Archimedean entropy functional. As a main consequence, we obtain a solution to the (uniform version of the) Yau-Tian-Donaldson Conjecture: a polarized smooth projective variety $(X,L)$ admits a cscK metric if and only if it is $\mathrm{Aut}^\circ(X,L)$-uniformly $K$-stable. This extends the known Yau-Tian-Donaldson correspondence for smooth Fano varieties.

math.AG↗

arXiv2025

Weighted extremal Kähler metrics on resolutions of singularities

Generalizing previous results of Arezzo-Pacard-Singer, Seyyedali-Székelyhidi and Hallam, we prove the invariance under smooth blowups of the class of weighted extremal Kähler manifolds, modulo a log-concavity assumption on the first weight. Through recent work of Di Nezza-Jubert-Lahdili and Han-Liu, this is obtained as a consequence of a general uniform coercivity estimate for the (relative, weighted) Mabuchi energy on the blowup, which applies more generally to any equivariant resolution of singularities of Fano type of a compact Kähler klt space whose Mabuchi energy is assumed to be coercive.

math.DG↗

arXiv2025

Singular cscK metrics on smoothable varieties

We prove the lower semi-continuity of the coercivity threshold of Mabuchi functional along a degenerate family of normal compact Kähler varieties with klt singularities. Moreover, we establish the existence of singular cscK metrics on $\mathbb{Q}$-Gorenstein smoothable klt varieties when the Mabuchi functional is coercive, these arise as a limit of cscK metrics on close-by fibres. The proof relies on developing a novel strong topology of pluripotential theory in families and establishing uniform estimates for cscK metrics.

math.CV↗

arXiv2024

Kähler-Einstein metrics on families of Fano varieties

Given a one-parameter family of $\mathbb{Q}$-Fano varieties such that the central fibre admits a unique Kähler-Einstein metric, we provide an analytic method to show that the neighboring fibre admits a unique Kähler-Einstein metric. Our results go beyond by establishing uniform a priori estimates on the Kähler-Einstein potentials along fully degenerate families of $\mathbb{Q}$-Fano varieties. In addition, we show the continuous variation of these Kähler-Einstein currents, and establish uniform Moser-Trudinger inequalities and uniform coercivity of the Ding functionals. Central to our article is introducing and studying a notion of convergence for quasi-plurisubharmonic functions within families of normal Kähler varieties. We show that the Monge-Ampère energy is upper semi-continuous with respect to this topology, and we establish a Demailly-Kollár result for functions with full Monge-Ampère mass.

math.CV↗

arXiv2024

Convexity of the Mabuchi functional in big cohomology classes

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.

math.DG↗

arXiv2024

Entropy for Monge-Ampère Measures in the Prescribed Singularities 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 $n=1$.

math.CV↗

arXiv2023

Kähler-Einstein metrics with positive curvature near an isolated log terminal singularity

We analyze the existence of Kähler-Einstein metrics of positive curvature in the neighborhood of a germ of a log terminal singularity $(X,p)$. This boils down to solve a Dirichlet problem for certain complex Monge-Ampère equations. We show that the solvability of the latter is independent of the shape of the domain and of the boundary data. We establish a Moser-Trudinger $(MT)_γ$ inequality in subcritical regimes $γ<γ_p$ and establish the existence of smooth solutions in that cases. We show that the expected critical exponent $\hatγ_p=\frac{n+1}{n} \widehat{\mathrm{vol}}(X,p)^{1/n}$ can be expressed in terms of the normalized volume, an important algebraic invariant of the singularity.

math.DG↗

arXiv2023

A relative Yau-Tian-Donaldson conjecture and stability thresholds

Generalizing Fujita-Odaka invariant, we define a function $\tildeδ$ on a set of generalized $b$-divisors over a smooth Fano variety. This allows us to provide a new characterization of uniform $K$-stability. A key role is played by a new Riemann-Zariski formalism for $K$-stability. For any generalized $b$-divisor $\mathbf{D}$, we introduce a (uniform) $\mathbf{D}$-log $K$-stability notion. We prove that the existence of a unique Kähler-Einstein metric with prescribed singularities implies this new $K$-stability notion when the prescribed singularities are given by the generalized $b$-divisor $\mathbf{D}$. We connect the existence of a unique Kähler-Einstein metric with prescribed singularities to a uniform $\mathbf{D}$-log Ding-stability notion which we introduce. We show that these conditions are satisfied exactly when $\tildeδ(\mathbf{D})>1$, extending to the $\mathbf{D}$-log setting the $δ$-valuative criterion of Fujita-Odaka and Blum-Jonsson. Finally we prove the strong openness of the uniform $\mathbf{D}$-log Ding stability as a consequence of the strong continuity of $\tildeδ$.

math.AG↗

arXiv2022

Quasi-monotone convergence of plurisubharmonic functions

The complex Monge-Ampère operator has been defined for locally bounded plurisubharmonic functions by Bedford-Taylor in the 80's. This definition has been extended to compact complex manifolds, and to various classes of mildly unbounded quasi-plurisubharmonic functions by various authors. As this operator is not continuous for the $L^{1}$-topology, several stronger topologies have been introduced over the last decades to remedy this, while maintaining efficient compactness criteria. The purpose of this note is to show that these stronger topologies are essentially equivalent to the natural quasi-monotone topology that we introduce and study here.

math.CV↗

arXiv2022

Kähler-Einstein metrics with prescribed singularities on Fano manifolds

Given a Fano manifold $(X,ω)$ we develop a variational approach to characterize analytically the existence of Kähler-Einstein metrics with prescribed singularities, assuming that these singularities can be approximated algebraically. Moreover, we define a function $α_ω$ on the set of prescribed singularities which generalizes Tian's $α$-invariant, showing that its upper level set $\{α_ω(\cdot)>\frac{n}{n+1}\}$ produces a subset of the Kähler-Einstein locus, i.e. of the locus given by all prescribed singularities that admit Kähler-Einstein metrics. In particular, we prove that many $K$-stable manifolds admit all possible Kähler-Einstein metrics with prescribed singularities. Conversely, we show that enough positivity of the $α$-invariant function at non-trivial prescribed singularities (or other conditions) implies the existence of genuine Kähler-Einstein metrics. Finally, through a continuity method, we also prove the strong continuity of Kähler-Einstein metrics on curves of totally ordered prescribed singularities when the relative automorphism groups are discrete.

math.DG↗

arXiv2022

Continuity method with movable singularities for classical Monge-Ampère equations

On a compact Kähler manifold $(X,ω)$, we study the strong continuity of solutions with prescribed singularities of complex Monge-Ampère equations with integrable Lebesgue densities. Moreover, we give sufficient conditions for the strong continuity of solutions when the right-hand sides are modified to include all (log) Kähler-Einstein metrics with prescribed singularities. Our findings can be interpreted as closedness of new continuity methods in which the densities vary together with the prescribed singularities. For Monge-Ampère equations of Fano type, we also prove an openness result when the singularities decrease. As an application, we deduce a strong stability result for (log-)Kähler Einstein metrics on semi-Kähler classes given as modifications of $\{ω\}$.

math.DG↗

arXiv2022

$L^{1}$ metric geometry of potentials with prescribed singularities on compact Kähler manifolds

Given $(X,ω)$ compact Kähler manifold and $ψ\in\mathcal{M}^{+}\subset PSH(X,ω)$ a model type envelope with non-zero mass, i.e. a fixed potential determing some singularities such that $\int_{X}(ω+dd^{c}ψ)^{n}>0$, we prove that the $ψ-$relative finite energy class $\mathcal{E}^{1}(X,ω,ψ)$ becomes a complete metric space if endowed with a distance $d$ which generalizes the well-known $d_{1}$ distance on the space of Kähler potentials. Moreover, for $\mathcal{A}\subset \mathcal{M}^{+}$ total ordered, we equip the set $X_{\mathcal{A}}:=\bigsqcup_{ψ\in\overline{\mathcal{A}}}\mathcal{E}^{1}(X,ω,ψ)$ with a natural distance $d_{\mathcal{A}}$ which coincides with the distance $d$ on $\mathcal{E}^{1}(X,ω,ψ)$ for any $ψ\in\overline{\mathcal{A}}$. We show that $\big(X_{\mathcal{A}},d_{\mathcal{A}}\big)$ is a complete metric space. As a consequence, assuming $ψ_{k}\searrow ψ$ and $ψ_{k},ψ\in \mathcal{M}^{+}$, we also prove that $\big(\mathcal{E}^{1}(X,ω,ψ_{k}),d\big)$ converges in a Gromov-Hausdorff sense to $\big(\mathcal{E}^{1}(X,ω,ψ),d\big)$ and that there exists a direct system $\Big\langle\big(\mathcal{E}^{1}(X,ω,ψ_{k}),d\big),P_{k,j}\Big\rangle$ in the category of metric spaces whose direct limit is dense into $\big(\mathcal{E}^{1}(X,ω,ψ),d\big)$.

math.DG↗

arXiv2022

The strong topology of $ω$-plurisubharmonic functions

On $(X,ω)$ compact Kähler manifold, given a model type envelope $ψ\in PSH(X,ω)$ (i.e. a singularity type) we prove that the Monge-Ampère operator is an homeomorphism between the set of $ψ$-relative finite energy potentials and the set of $ψ$-relative energy measures endowed with their strong topologies given as the coarsest refinements of the weak topologies such that the relative energies become continuous. Moreover, given a totally ordered family $\mathcal{A}$ of model type envelopes with positive total mass representing different singularities types, the sets $X_{\mathcal{A}}, Y_{\mathcal{A}}$ given respectively as the union of all $ψ$-relative finite energy potentials and of all $ψ$-relative finite energy measures varying $ψ\in\overline{\mathcal{A}}$ have two natural strong topologies which extends the strong topologies on each component of the unions. We show that the Monge-Ampère operator produces an homeomorphism between $X_{\mathcal{A}}$ and $Y_{\mathcal{A}}$. As an application we also prove the strong stability of a sequence of solutions of prescribed complex Monge-Ampère equations when the measures have uniformly $L^{p}$-bounded densities for $p>1$ and the prescribed singularities are totally ordered.

math.DG↗

arXiv2021

Multipoint Okounkov bodies

Starting from the data of a big line bundle $L$ on a projective manifold $X$ with a choice of $N\geq 1$ different points on $X$ we provide a new construction of $N$ Okounkov bodies which encodes important geometric features of ($L\to X,p_{1},\dots,p_{N}$) such as the volume of $L$, the (moving) multipoint Seshadri constant of $L$ at $p_{1},\dots,p_{N}$, and the possibility to construct Kähler packings centered at $p_{1},\dots,p_{N}$. Toric manifolds and surfaces are examined in detail.

math.AG↗