Research
Publications and Preprints
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Demailly-Lelong 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 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.
We also provide an estimate for quotient singularities and sharp estimates for two-dimensional ADE singularities.
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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.
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Kähler-Einstein 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 Kähler-Einstein metrics and establish uniform a priori estimates on the Kähler-Einstein potentials along degenerate families of Q-Fano varieties.
Moreover, we show that these Kähler-Einstein currents vary continuously, and we prove uniform Moser-Trudinger inequalities.
The core of the article regards a notion of convergence of quasi-plurisubharmonic functions in families of normal Kähler varieties that we introduce and study here.
We show that the Monge-Ampère energy is upper semi-continuous with respect to this convergence, and we establish a Demailly-Kollár result for functions with full Monge-Ampère mass.
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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.
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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
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Familles de métriques hermitiennes canoniques
[text] PhD thesis supervised by V. Guedj and H. Guenancia and defended on June 19, 2023
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Regularity of geodesics in the space of Kähler metrics
[text]
M2 report, defended at Université Paul Sabatier in July 2020