Research

Publications and Preprints

1. Gauduchon metrics and Hermite-Einstein metrics on non-Kähler varieties
   [abstract] [arXiv]
   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.

2. Weighted cscK metrics on Kähler varieties (with T. D. Tô)
   [abstract] [arXiv] submitted.
   We study the weighted constant scalar curvature Kähler equations on mildly singular Kähler varieties. Under an assumption on the existence of certain resolution of singularities, we prove the existence of singular weighted constant scalar curvature Kähler metrics when the weighted Mabuchi functional is coercive. This extends the work of Chen and Cheng to the singular weighted setting.

3. Demailly-Lelong numbers on complex spaces
   [abstract] [arXiv] [journal] Math. Z. 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.

4. 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 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.

5. Kähler-Einstein metrics on families of Fano varieties (with A. Trusiani)
   [abstract] [arXiv] [journal] J. Reine Angew. Math. (Crelle's Journal) 819 (2025), 45-87.
   See also Oberwolfach Report No. 29/2023 Workshop "Differentialgeometrie im Großen"
   Given a one-parameter family of 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 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.

6. Families of singular Chern-Ricci flat metrics
   [abstract] [arXiv] [journal] J. Geom. Anal. 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.

7. Singular Gauduchon metrics
   [abstract] [arXiv] [journal] Compos. Math. 158 (2022), no. 6, 1314-1328.
   In 1977, Gauduchon proved that on every compact hermitian manifold $(X,\omega)$ there exists a conformally equivalent hermitian metric $\omega_G$ which satisfies $dd^c \omega_G^{n-1} = 0$. 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
  
• 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
• Geometry and TACoS
• GdT Complexe CIRGET

Last modified: March 05, 2025