Carlo Scarpa, Université Lyon 1

I conduct research in Differential Geometry at the Institut Camille Jordan (Université Lyon 1) as a Marie Skłodowska-Curie postdoctoral fellow, working on the project CR-Yamabe problem, Kähler metrics and K-stability with Eveline Legendre.

Previously, I worked in the CIRGET laboratory of UQAM with Julien Keller and at SISSA and IGAP with Jacopo Stoppa.

You can find a brief CV (not always up to date) here.

Primary email address: scarpa@math.univ-lyon1.fr.

My research focus is on Kähler geometry, particularly on the existence of Kähler metrics with special curvature properties, such as (in increasing generality) Kähler-Einstein metrics, constant scalar curvature (cscK) metrics or extremal Kähler metrics.

For an introduction to these topics I recommend this book by Gábor Székelyhidi, Paul Gauduchon's book, and two papers by Richard Thomas and Julius Ross and Ruadhaí Dervan.

I am mainly interested in problems where the existence of metrics with special curvature properties is coupled with other aspects of the theory of complex manifolds, like the study of deformations of complex structures.

Announcements:

there will be a Geometric Analysis workshop in Lyon in October 2025, follow the link for more information. We have limited space and founding, let me know in case you are interested in participating!

Research

All my preprints can be found on the arxiv.

Papers

The CR Yamabe invariant and constant scalar curvature Sasaki metrics, with Abdellah Lahdili and Eveline Legendre. arxiv preprint arXiv:2509.00743[math.DG].
Polynomial stability conditions for vector bundles: positivity, equivariance and blow-ups, with Rémi Delloque, Achim Napame and Carl Tipler. arxiv preprint arXiv:2506.23842 [math.AG].
Z-critical equations for holomorphic vector bundles on Kähler surfaces, with Julien Keller. arxiv preprint arxiv:2405.03312[math.DG].
The Einstein-Hilbert functional and the Donaldson-Futaki invariant, with Abdellah Lahdili and Eveline Legendre. arxiv preprint arXiv:2310.11625[math.DG].
A K-energy functional for complexified Kähler classes. arxiv preprint arXiv:2307.09904[math.DG].
Special representatives of complexified Kähler classes, with Jacopo Stoppa. Selecta Mathematica 30, 64 (2024), https://doi.org/10.1007/s00029-024-00955-1
Scalar curvature and deformations of complex structures. Journal für die reine und angewandte Mathematik (Crelles Journal) 797, 255-283 (2023), https://doi.org/10.1515/crelle-2023-0010
The HcscK system in symplectic coordinates, with Jacopo Stoppa. Mathematische Zeitschrift 301, 75–113 (2022), https://doi.org/10.1007/s00209-021-02902-8.
Solutions to Donaldson’s hyperkähler reduction on a curve, with Jacopo Stoppa. Journal of Geometric Analysis 31, 2871–2889 (2021), https://doi.org/10.1007/s12220-020-00377-3.
Scalar curvature and an infinite-dimensional hyperkähler reduction, with Jacopo Stoppa. Asian Journal of Mathematics, Vol 24, N. 4, 671–724 (2020), https://dx.doi.org/10.4310/AJM.2020.v24.n4.a7.

My thesis:

the Hitchin-cscK system

Teaching

In the Fall term of 2023 I taught the course MATH-139 Calculus 1 with Precalculus at McGill University. All information and student resources can be found in the course's page on McGill's myCourses platform. My notes and the lectures overview are available to students on myCourses.

Moment maps in differential geometry

These are some notes (handwritten, unfortunately) of a graduate course I taught in SISSA in the 2022 Spring term. I will update a revised TeX version at some point in the future.

course presentation: lect0
symplectic forms: lect1
some Kähler geometry: lect2
Hamiltonian actions: lect3
symplectic quotients: lect4
algebraic stability: lect5
the Kempf-Ness Theorem: lect6
Hermite-Einstein bundles: lect7
slope stability and Einstein metrics: lect8
scalar curvature as a moment map: lect9
quantization of the cscK problem: lect10

Some notes:

Others:

together with Eveline Legendre and Rémi Reboulet, I have organised a workshop on Kähler and non-Archimedean Geometry in Lyon in November 2024. You can find more information on the conference website.

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