arXiv subjects

Simon Jubert

Publications and source records attributed to Simon Jubert.

8 recordsLinked to original sources

arXiv2026

New extremal K\"ahler metrics on projective bundles

Consider a holomorphic vector bundle $E$ over a compact complex curve $C$ which decomposes as a sum of stable vector bundles. For the projectivization $\mathbb{P}(E)$, we prove that the existence of a compatible extremal almost K\"ahler (aK) metric of involutive type in the sense of Lejmi is equivalent to the existence of a Calabi extremal K\"ahler metric. This result rests on the Yau--Tian--Donaldson correspondence in terms of the moment polytope $\Delta$ for $\mathbb{P}(E)$, proved by the author and Yin in a previous work. The main advantage is that compatible extremal aK metrics of involutive type are solutions to a second-order linear PDE, rather than a fourth-order nonlinear PDE for Calabi's extremal K\"ahler metrics. As an application, we prove that when $E$ has rank $4$ and $C$ is an elliptic curve or the projective line, $\mathbb{P}(E)$ is a Calabi dream manifold, i.e. admits an extremal K\"ahler metric in every K\"ahler class.

math.DG↗

arXiv2026

Relative uniform Yau--Tian--Donaldson correspondence for projective bundles over a curve

This paper is concerned with a relative uniform Yau--Tian--Donaldson correspondence, in terms of test configurations, for the projectivization $ \mathbb{P}(E) $ of a holomorphic vector bundle $ E $ over a smooth curve. For any K\"ahler class $ [\omega] $ on $ \mathbb{P}(E) $, we construct K\"ahler test configurations, which we call \emph{compatible test configurations}. They are obtained by gluing horospherical test configurations from the fibers, arising from convex functions on a suitable moment polytope $ \Delta $ following the construction of Delcroix, to the principal bundle associated with $ \mathbb{P}(E) $. Using the generalized Calabi ansatz of Apostolov--Calderbank--Gauduchon--T{\o}nnesen-Friedman on these test configurations, we show that the relative uniform stability of $ (\mathbb{P}(E),[\omega]) $ for compatible test configurations implies the existence of an extremal metric in this class, thereby establishing the equivalence. Along the way, we prove that these two conditions are equivalent to the weighted uniform stability of $ \Delta $ for suitable explicit weight functions defined from the topological data of $ \mathbb{P}(E) $.

math.DG↗

arXiv2025

Weighted cscK metric (II): the continuity method

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 $X$.

math.DG↗

arXiv2025

Numerical invariants for weighted cscK metrics

In K-stability, the delta invariant of a Fano variety encodes the existence of K\"ahler-Einstein metrics. We introduce a weighted analytic delta invariant, and a reduced version, that characterize the existence of weighted solitons. We further prove a sufficient condition of existence of weighted cscK metrics in terms of this invariant. We elucidate the relation between the weighted delta invariant and the greatest lower bound on the weighted Ricci curvature, called the weighted beta invariant. We provide a general upper bound for the weighted beta invariant in terms of moment images. Finally, we investigate how the geometry of semisimple principal fibrations, whose basis is not assumed to be cscK, allows to estimate their beta invariant in terms of the basis and the weighted fiber. Most of our statements are new even in the trivial weights settings, that is, for K\"ahler-Einstein and cscK metrics.

math.DG↗

arXiv2024

Weighted cscK metric (I): a priori estimates

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.

math.DG↗

arXiv2022

An effective weighted K-stability condition for polytopes and semisimple principal toric fibratons

The second author has shown that existence of extremal K\"ahler metrics on semisimple principal toric fibrations is equivalent to a notion of weighted uniform K-stability, read off from the moment polytope. The purpose of this article is to prove various sufficient conditions of weighted uniform K-stability which can be checked effectively and explore the low dimensional new examples of extremal K\"ahler metrics it provides.

math.DG↗

arXiv2021

A Yau-Tian-Donaldson correspondence on a class of toric fibrations

We established a Yau--Tian--Donaldson type correspondence, expressed in terms of a single Delzant polytope, concerning the existence of extremal K\"ahler metrics on a large class of toric fibrations, introduced by Apostolov--Calderbank--Gauduchon--Tonnesen-Friedman and called semi-simple principal toric fibrations. We use that an extremal metric on the total space corresponds to a weighted constant scalar curvature K\"ahler metric (in the sense of Lahdili) on the corresponding toric fiber in order to obtain an equivalence between the existence of extremal K\"ahler metrics on the total space and a suitable notion of weighted uniform K-stability of the corresponding Delzant polytope. As an application, we show that the projective plane bundle $\mathbb{P}(\mathcal{L}_0\oplus\mathcal{L}_1 \oplus \mathcal{L}_2)$, where $\mathcal{L}_i$ are holomorphic line bundles over an elliptic curve, admits an extremal metric in every K\"ahler class.

math.DG↗

arXiv2021

Weighted K-stability and coercivity with applications to extremal Kahler and Sasaki metrics

We show that a compact weighted extremal Kahler manifold (as defined by the third named author) has coercive weighted Mabuchi energy with respect to a maximal complex torus in the reduced group of complex automorphisms. This provides a vast extension and a unification of a number of results concerning Kahler metrics satisfying special curvature conditions, including constant scalar curvature Kahler metrics, extremal Kahler metrics, Kahler-Ricci solitons and their weighted extensions. Our result implies the strict positivity of the weighted Donaldson-Futaki invariant of any non-product equivariant smooth K\"ahler test configuration with reduced central fibre, a property also known as weighted K-polystability on such test configurations. For a class of fibre-bundles, we use our result in conjunction with the recent results of Chen-Cheng, He, and Han-Li in order to characterize the existence of extremal Kahler metrics and Calabi-Yau cones associated to the total space, in terms of the coercivity of the weighted Mabuchi energy of the fibre. In particular, this yields an existence result for Sasaki-Einstein metrics on Fano toric fibrations, extending the results of Futaki-Ono-Wang in the toric Fano case, and of Mabuchi-Nakagawa in the case of Fano projective line bundles.

math.DG↗