arXiv subjects

Rémi Reboulet

Publications and source records attributed to Rémi Reboulet.

12 recordsLinked to original sources

arXiv2026

A birational version of K-stability for big classes

We introduce a theory of uniform K-stability for big line bundles on smooth projective varieties. This extends the existing theory both for varieties with ample line bundles, and for varieties with big anticanonical class. Our main result gives a valuative characterisation of uniform K-stability, through finite collections of divisorial valuations. We further prove that uniform K-stability is preserved under pullbacks and certain pushforwards, which implies that uniform K-stability is well-defined at the level of b-divisors.

math.AG↗

arXiv2026

Filtrations and asymptotic geometry of non-Archimedean norms on section rings

This article is concerned with the metric study of a construction of Gérardin of the action of the boundary at infinity of the space of norms on a non-Archimedean vector space, and its generalisation to graded algebras. Namely, given (X,L) a polarised variety over an arbitrary non-Archimedean field, we show that there is a jointly d_1-contracting action of the space of filtrations of the section ring R(X,L) on the space of graded norms on R(X,L). This naturally yields non-Archimedean geodesic rays and infinite-dimensional flats in this setting, generalising previous work of the author and Witt Nyström. It is further shown that relative limit measures converge along geodesic rays, providing a result on the d_p-radial geometry of graded norms, analogous to a recent result of Finski in the Archimedean case.

math.AG↗

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

Flats in the space of Kähler metrics and Okounkov bodies

We study sufficient conditions for the existence of flat subspaces in the space of continuous plurisubharmonic metrics on a polarised complex projective manifold, relying on the generalised Legendre transform to the Okounkov body defined by Witt Nystrom, and a result of Schwer--Lytchak.

math.DG↗

arXiv2025

Infinite-dimensional flats in the space of positive metrics on an ample line bundle

We show that any continuous positive metric on an ample line bundle L lies at the apex of many infinite-dimensional Mabuchi-flat cones. More precisely, given any bounded graded filtration F of the section ring of L, the set of bounded decreasing convex functions on the support of the Duistermaat--Heckman measure of F embeds L^p-isometrically into the space of bounded positive metrics on L with respect to Darvas' d_p distance for all p\in[1,\infty), and in particular with respect to the Mabuchi metric (p=2).

math.DG↗

arXiv2025

A Hilbert--Mumford criterion for generalised Monge--Ampère equations

We give a new numerical criterion in the spirit of GIT for existence of solutions to inverse Hessian equations, including in particular the J-equation. Our criterion is formulated in terms of stability of pairs in the sense of Paul. To that end, we build on previous work of the author with Dervan, and generalise a result of Zhang, proving isometry between generalised Chow line bundles and mixed Deligne pairings.

math.DG↗

arXiv2025

The birational geometry of GIT quotients

Geometric Invariant Theory (GIT) produces quotients of algebraic varieties by reductive groups. If the variety is projective, this quotient depends on a choice of polarisation; by work of Dolgachev-Hu and Thaddeus, it is known that two quotients of the same variety using different polarisations are related by birational transformations. Only finitely many birational varieties arise in this way: variation of GIT fails to capture the entirety of the birational geometry of GIT quotients. We construct a space parametrising all possible GIT quotients of all birational models of the variety in a simple and natural way, which captures the entirety of the birational geometry of GIT quotients in a precise sense. It yields in particular a compactification of a birational analogue of the set of stable orbits of the variety.

math.AG↗

arXiv2024

Arcs, stability of pairs and the Mabuchi functional

We prove various results involving arcs - which generalise test configurations - within the theory of K-stability. Our main result characterises coercivity of the Mabuchi functional on spaces of Fubini-Study metrics in terms of uniform K-polystability with respect to arcs, thereby proving a version of a conjecture of Tian. The main new tool is an arc version of a numerical criterion for Paul's theory of stability of pairs, for which we also provide a suitable generalisation applicable to pairs with nontrivial stabiliser. We give two applications. Firstly, we give a new proof of a version of the Yau-Tian-Donaldson conjecture for Fano manifolds, along the lines originally envisaged by Tian - allowing us to reduce the general Yau-Tian-Donaldson conjecture to an analogue of the partial C^0-estimate. Secondly, for a (possibly singular) polarised variety which is uniformly K-polystable with respect to arcs, we show that the associated Cartan subgroup of its automorphism group is reductive. In particular, uniform K-stability with respect to arcs implies finiteness of the automorphism group. This generalises work of Blum-Xu for Fano varieties.

math.AG↗

arXiv2024

Ding stability and Kähler-Einstein metrics on manifolds with big anticanonical class

We introduce a notion of uniform Ding stability for a projective manifold with big anticanonical class, and prove that the existence of a unique Kähler-Einstein metric on such a manifold implies uniform Ding stability. The main new techniques are to develop a general theory of Deligne functionals - and corresponding slope formulas - for singular metrics, and to prove a slope formula for the Ding functional in the big setting. This extends work of Berman in the Fano situation, when the anticanonical class is actually ample, and proves one direction of the analogue of the Yau-Tian-Donaldson conjecture in this setting. We also speculate about the relevance of uniform Ding stability and K-stability to moduli in the big setting.

math.DG↗

arXiv2021

The space of finite-energy metrics over a degeneration of complex manifolds

Given a degeneration of compact projective complex manifolds $X$ over the punctured disc, with meromorphic singularities, and a relatively ample line bundle $L$ on $X$, we study spaces of plurisubharmonic metrics on $L$, with particular focus on (relative) finite-energy conditions. We endow the space $\hat \cE^1(L)$ of relatively maximal, relative finite-energy metrics with a $d_1$-type distance given by the Lelong number at zero of the collection of fibrewise Darvas $d_1$-distances. We show that this metric structure is complete and geodesic. Seeing $X$ and $L$ as schemes $X_\K$, $L_\K$ over the discretely-valued field $\K=\mathbb{C}((t))$ of complex Laurent series, we show that the space $\cE^1(L_\K\an)$ of non-Archimedean finite-energy metrics over $L_\K\an$ embeds isometrically and geodesically into $\hat \cE^1(L)$, and characterize its image. This generalizes previous work of Berman-Boucksom-Jonsson, treating the trivially-valued case. We investigate consequences regarding convexity of non-Archimedean functionals.

math.DG↗

arXiv2020

Plurisubharmonic geodesics in spaces of non-Archimedean metrics of finite energy

Given a polarized projective variety (X,L) over any non-Archimedean field, assuming continuity of envelopes, we show that the space of finite-energy metrics on L is a geodesic metric space, where geodesics are given as maximal psh segments. Given two continuous psh metrics, we show that the maximal segment joining them is furthermore continuous.

math.AG↗

arXiv2020

The asymptotic Fubini-Study operator over general non-Archimedean fields

Given an ample line bundle $L$ over a projective $\mathbb{K}$-variety $X$, with $\mathbb{K}$ a non-Archimedean field, we study limits of non-Archimedean metrics on $L$ associated to submultiplicative sequences of norms on the graded pieces of the section ring $R(X,L)$. We show that in a rather general case, the corresponding asymptotic Fubini-Study operator yields a one-to-one correspondence between equivalence classes of bounded graded norms and bounded plurisubharmonic metrics that are regularizable from below. This generalizes results of Boucksom-Jonsson where this problem has been studied in the trivially valued case.

math.AG↗