Research
Publications and Preprints

DemaillyLelong numbers on complex spaces
[abstract]
[arXiv] submitted.
We establish a pointwise comparison of two notions of Lelong numbers of plurisubharmonic functions defined on singular complex spaces.
This shows a conjecture proposed by BermanBoucksomEyssidieuxGuedjZeriahi, affirming that the DemaillyLelong 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.
We also provide an estimate for quotient singularities and sharp estimates for twodimensional ADE singularities.

Singular cscK metrics on smoothable varieties (with T. D. Tô and A. Trusiani)
[abstract]
[arXiv] submitted.
We prove the lower semicontinuity 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 QGorenstein smoothable klt varieties when the Mabuchi functional is coercive, these arise as a limit of cscK metrics on closeby fibres.
The proof relies on developing a novel strong topology of pluripotential theory in families and establishing uniform estimates for cscK metrics.

KählerEinstein metrics on families of Fano varieties (with A. Trusiani)
[abstract]
[arXiv] submitted.
See also Oberwolfach Report: No. 29/2023 (Differentialgeometrie im Großen)
We provide an analytic proof of the openness of the existence of unique KählerEinstein metrics and establish uniform a priori estimates on the KählerEinstein potentials along degenerate families of QFano varieties.
Moreover, we show that these KählerEinstein currents vary continuously, and we prove uniform MoserTrudinger inequalities.
The core of the article regards a notion of convergence of quasiplurisubharmonic functions in families of normal Kähler varieties that we introduce and study here.
We show that the MongeAmpère energy is upper semicontinuous with respect to this convergence, and we establish a DemaillyKollár result for functions with full MongeAmpère mass.

Families of singular ChernRicci 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 MongeAmpère equations on a family of hermitian varieties.
This generalizes a theorem of Di NezzaGuedjGuenancia to hermitian contexts.
The main result can be applied to study the uniform boundedness of ChernRicci flat potentials in conifold transitions.

Singular Gauduchon metrics
[abstract]
[arXiv]
[journal]
Compos. Math. 158 (2022), no. 6, 13141328.
In 1977, Gauduchon proved that on every compact hermitian manifold $(X,\omega)$ there exists a conformally equivalent hermitian metric $\omega_G$ which satisfies $dd^c \omega_G^{n1} = 0$.
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