Research

Publications and Preprints

1. From Calabi's extremal metrics to scalar-flat Kähler cones (with V. Apostolov and A. Lahdili)
   [abstract] [arXiv] Submitted.
   We prove that for any smooth polarized complex ${\small n}$-dimensional manifold ${\small (X, L_X)}$ which admits an extremal Kähler metric in ${\small c_1(L_X)}$, and for any integer ${\small k}$ large enough (in terms of a bound depending on ${\small (X, L_X)}$), the ${\small (n+k+1)}$-dimensional complex cone associated with the polarization ${\small L_X \otimes \mathcal{O}_{\mathbb{P}^k}(1)}$ over ${\small X \times \mathbb{P}^k}$ admits a scalar-flat Kähler cone metric. Equivalently, the unweighted Sasaki join of a smooth compact quasi-regular extremal Sasaki manifold with a regular Sasaki sphere ${\small \mathbb{S}^{2k+1}}$ of sufficiently large dimension ${\small (2k+1)}$ admits a Sasaki metric of constant (positive) scalar curvature. This gives an affirmative answer to an asymptotic version of a question raised by Boyer--Huang--Legendre--Tønnesen-Friedman in [BHLTF21].
2. A note on orbifold regularity of canonical metrics (with H. Guenancia and M. Păun)
   [abstract] [arXiv] Submitted.
   In this short note, we prove that on a compact Kähler variety ${\small X}$ with log terminal singularities and ${\small c_1(X)=0}$, any singular Ricci-flat Kähler metric has orbifold singularities in restriction to the orbifold locus of ${\small X}$.
3. Gauduchon metrics and Hermite-Einstein metrics on non-Kähler varieties
   [abstract] [arXiv] Submitted.
   We show the existence of Gauduchon metrics on arbitrary compact hermitian varieties, generalizing our previous work on smoothable singularities. These metrics allow us to define the notion of slope stability for torsion-free coherent sheaves on compact normal varieties that are not necessarily Kähler. Then we prove the existence and uniqueness of singular Hermite-Einstein metrics for slope-stable reflexive sheaves on non-Kähler normal varieties.
4. Weighted cscK metrics on Kähler varieties (with T. D. Tô)
   [abstract] [arXiv] [journal] Journal of the London Mathematical Society 113 (2026), no. 2, Paper No. e70442.
   We study the weighted constant scalar curvature Kähler equations on mildly singular Kähler varieties. Assuming the existence of a suitable resolution of singularities, we establish the existence of singular weighted cscK metrics when the weighted Mabuchi functional is coercive for an extremal weight. This extends the works of Chen-Cheng and He to the singular weighted setting. Moreover, we provide a method for constructing examples of singular cscK metrics inspired by the work of Arezzo-Pacard. In contrast to the usual gluing techniques, our approach does not require a precise understanding about of the metric behavior near the singular locus.
5. Demailly-Lelong numbers on complex spaces
   [abstract] [arXiv] [journal] Mathematische Zeitschrift 309 (2025), no. 1, Paper No. 4, 17 pp.
   We prove a conjecture proposed by Berman-Boucksom-Eyssidieux-Guedj-Zeriahi, affirming that the Demailly-Lelong number can be determined through a combination of intersection numbers given by the divisorial part of the potential and the SNC divisors over a log resolution of the maximal ideal of a given point. Moreover, this result establishes a pointwise comparison of two different notions of Lelong numbers of plurisubharmonic functions defined on singular complex spaces. We also provide an estimate for quotient singularities and sharp estimates for two-dimensional ADE singularities.
6. Singular cscK metrics on smoothable varieties (with T. D. Tô and A. Trusiani)
   [abstract] [arXiv] Submitted.
   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 ${\small \mathbb{Q}}$-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.
7. Kähler-Einstein metrics on families of Fano varieties (with A. Trusiani)
   [abstract] [arXiv] [journal] Journal für die reine und angewandte Mathematik (Crelle's Journal) 819 (2025), 45-87.
   See also Oberwolfach Report No. 29/2023 Workshop "Differentialgeometrie im Großen"
   Given a one-parameter family of ${\small \mathbb{Q}}$-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 ${\small \mathbb{Q}}$-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.
8. Families of singular Chern-Ricci flat metrics
   [abstract] [arXiv] [journal] The Journal of Geometric Analysis 33 (2023), no. 2, Paper No. 66, 32 pp.
   We prove uniform a priori estimates for degenerate complex Monge-Ampère equations on a family of hermitian varieties. This generalizes a theorem of Di Nezza-Guedj-Guenancia to hermitian contexts. The main result can be applied to study the uniform boundedness of Chern-Ricci flat potentials in conifold transitions.
9. Singular Gauduchon metrics
   [abstract] [arXiv] [journal] Compositio Mathematica 158 (2022), no. 6, 1314-1328.
   In 1977, Gauduchon proved that on every compact hermitian manifold \({\small (X,\omega)}\) there exists a conformally equivalent hermitian metric \({\small \omega_G}\) which satisfies \({\small dd^c \omega_G^{n-1}= 0}\). In this note, we extend this result to irreducible compact singular hermitian varieties which admit a smoothing.

Miscellaneous

• Familles de métriques hermitiennes canoniques
  [text] PhD thesis supervised by V. Guedj and H. Guenancia and defended on June 19, 2023
  Jury : T. C. Collins (Rapporteur), P. Eyssidieux (Examinateur), A. Fino (Rapportrice), V. Guedj (Directeur de thèse), H. Guenancia (Directeur de thèse), M. Toma (Président du jury)
  Dans cette thèse, nous nous intéressons aux métriques singulières canoniques et spéciales sur une famille de variétés compactes complexes possiblement non-kählériennes.
  
  Le premier résultat concerne l'existence de métriques de Gauduchon "singulières". Sous certaines hypothèses sur les singularités, nous prouvons qu'une telle variété singulière compacte admet une métrique de Gauduchon bornée dans la classe conforme d'une métrique hermitienne lisse fixée. Ceci généralise le théorème de Gauduchon dans le cas singulier. D'autre part, dans chaque classe conforme, nous montrons également l'unicité des métriques de Gauduchon bornées à une constante multiplicative près. De plus, nous obtenons une propriété d'extension de la \((n-1)\)ème puissance d'une métrique de Gauduchon bornée comme courant plurifermé positif sur une variété singulière.
  
  Dans la deuxième partie, nous cherchons à établir des estimations uniformes pour les solutions d'équations de Monge-Ampère complexes pour une famille de variétés hermitiennes. Ensuite, nous montrons une estimation uniforme $L^1$ des fonctions quasi-plurisousharmoniques sup-normalisées quand la famille est un lissage d'une variété hermitienne qui n'a que des singularités isolées. La preuve s'appuie sur la théorie du pluripotentiel et un contrôle uniforme des facteurs conformes de Gauduchon. Combinée à l'estimation uniforme $L^p$ des densités canoniques, nous obtenons un contrôle uniforme des solutions des familles d'équations de Monge-Ampère complexes dans de tels cas.
• Regularity of geodesics in the space of Kähler metrics
  [text] M2 report, defended at Université Paul Sabatier in July 2020

Some links

• My Google scholar, arXiv page
• GdT Complexe CIRGET
• Geometry and TACoS

Last modified: April 16, 2026