Research

Publications and Preprints

1. 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.

2. 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.

3. Kähler-Einstein metrics on families of Fano varieties (with A. Trusiani)
   [abstract] [arXiv] [journal] J. Reine Angew. Math. (Crelle's Journal), published online.
   See also Oberwolfach Report: No. 29/2023 (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.

4. 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.

5. 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

Last modified: November 18, 2024