RESEARCH INTERESTS
My research lies at the interface of complex differential geometry, algebraic geometry, and geometric analysis. I am especially interested in canonical metrics — including Kähler–Einstein, cscK, special non-Kähler metrics — and, in particular, in the geometric structures that arise from their degenerations.
A recurring theme in my work is that analytic phenomena in limits of canonical metrics often have algebraic counterparts. This can lead to two interrelated outcomes:
construction of new canonical algebraic structures associated with moduli spaces and singularities —such as compact K-moduli spaces of Fano varieties and refined invariants of singularities— obtained by incorporating input from geometric PDEs into the algebro-geometric framework;
the emergence of an “algebraic computability” of highly non-explicit differential-geometric objects —such as Einstein metrics— through explicit descriptions of their degenerations via concrete algebraic tools.
This viewpoint underlies several of my works, including first constructions of compactified Kähler-Einstein/K-moduli spaces of Fano varieties in different contexts; constructions and regularity results for Calabi-Yau metrics singular along divisors and their relations with the Bogomolov-Miyaoka-Yau inequality; algebraic descriptions of multiscale bubbling phenomena in degenerations of Kähler-Einstein metrics; and gluing and adiabatic constructions for canonical metrics in both Kähler and non-Kähler geometry.
A further direction of my work concerns special metrics in non-Kähler geometry, where the appropriate notions of canonical metrics are still less understood. In particular, this has led me to study Yamabe-type problems and infinite-dimensional moment-map-type equations in this setting.
For a general introduction to the scientific area, see Wikipedia page on Complex Geometry.
Most of my publications are freely available on ArXiv.
Publications of my Research Group (PhDs and Postdocs), under Villum funded activities, can be found here: projects.au.dk/cmcg/publications.
Past collaborators: D. Angella, C. Arezzo, M. de Borbon, S. Calamai, P. Gallardo, F. Giusti, J. Martinez-Garcia, Y. Odaka, F. Pediconi, L.M. Sektnan, C. Scarpa, S. Sun, J. Windare, C. Yao.
Mentors: C. Arezzo, O. Biquard, S. K. Donaldson, J. Ross.
PRE-PRINTS
D. Angella, F. Pediconi, C. Scarpa, C. Spotti, J. Windare, Yamabe-type problems on compact Hermitian manifolds,
Pre-print: arXiv:2605.31028
We introduce and study a one-parameter Hermitian deformation of the Yamabe problem on compact complex manifolds. The deformation is defined by adding natural torsion terms to the Riemannian scalar curvature, and includes both the classical Yamabe equation and a scalar-curvature equation arising in locally conformally Kähler geometry as a momentum map. We analyze criteria for the existence of solutions, and discuss several examples.
M. de Borbon, C. Spotti, SNC Kähler-Einstein metrics and RCD spaces,
Pre-print: arXiv:2601.1874
We show that Kähler-Einstein metrics with cone singularities along simple normal crossing (SNC) divisors define RCD spaces, both in the compact setting and in certain non-compact cases, thereby producing many examples of Einstein RCD spaces. In particular, we show the existence of smooth non-compact 4-manifolds carrying ALE Ricci-flat RCD(0,4) metrics with any space form S3/Γ as the link of the tangent cone at infinity, answering a question raised by D. Semola. Our proofs rely on the characterization of RCD spaces in the almost-smooth setting due to S. Honda and Honda-Sun.
RESEARCH PUBLICATIONS
C. Spotti, On multiscale aspects of Kähler-Einstein metrics and algebraic geometry,
Accepted on EMS Surv. Math. Sci. 2025 Arxiv.
We discuss how metric limits and rescalings of Kähler-Einstein metrics connect with Algebraic Geometry, mostly in relation to the study of moduli spaces of varieties, and singularities. Along the way, we describe some elementary examples, review some recent results, and propose some tentative conjectural pictures.
F. Giusti, C. Spotti, Chern-Ricci flat balanced metrics on small resolutions of Calabi-Yau threefolds,
IMRN, Issue 18, (2025). Arxiv.
Given a (smoothable) projective nodal Kähler Calabi-Yau threefold, we show, via a gluing construction, that all its - possibly non-Kähler - small resolutions admit Chern-Ricci flat balanced metrics, which among other things solve the dilatino equation appearing in the Hull-Strominger system.
F. Giusti, C. Spotti, A Kummer construction for Chern-Ricci flat balanced manifolds,
Math Z., Volume 308, number 61, (2024). Arxiv.
Given a non-Kähler Calabi-Yau orbifold with a finite family of isolated singularities endowed with a Chern-Ricci flat balanced metric, we show, via a gluing construction, that all its crepant resolutions admit Chern-Ricci flat balanced metrics, and discuss applications to the search of solutions for the Hull-Strominger system. We also describe the scenario of singular threefolds with ordinary double points, and see that similarly is possible to obtain balanced approximately Chern-Ricci flat metrics.
M. de Borbon, C. Spotti, Some models for bubbling of (log) Kähler-Einstein metrics,
Ann. Univ. Ferrara (special Edge volume), Volume 70, pages 1037–1068, (2024). Arxiv.
We investigate aspects of the metric bubble tree for non-collapsing degenerations of (log) Kähler-Einstein metrics in complex dimensions one and two, and further describe a conjectural higher dimensional picture.
L. M. Sektnan, C. Spotti, Extremal metrics on the total space of destabilizing test configurations,
Math. Ann. Volume 388, pages 3427–3461, (2024). Arxiv.
We construct extremal metrics on the total space of certain destabilising test configurations for strictly semistable Kähler manifolds. This produces infinitely many new examples of manifolds admitting extremal Kähler metrics. It also shows for such metrics a new phenomenon of jumping of the complex structure along fibres.
M. de Borbon, C. Spotti, Calabi-Yau metrics with conical singularities along hyperplane arrangements,
J. Diff. Geom. Vol. 123, Issue 2, (2023) , pag 195-239. Arxiv.
Given a weighted line arrangement in the projective plane, with weights satisfying natural constraint conditions, we show the existence of a Ricci-flat Kähler metric with cone singularities along the lines asymptotic to a polyhedral Kähler cone at each multiple point. Moreover, we discuss a Chern-Weil formula that expresses the energy of the metric as a `logarithmic' Euler characteristic with points weighted according to the volume density of the metric.
D. Angella, S. Calamai. F. Pediconi, C. Spotti, A moment map for twisted-Hamiltonian vector fields on locally conformally Kähler manifolds,
Transform. Groups (2023). Arxiv.
We extend the classical Donaldson-Fujiki interpretation of the scalar curvature as moment map in Kähler Geometry to the wider framework of locally conformally Kähler Geometry.
J. Martinez-Garcia, C. Spotti, Some observations on the dimension of Fano K-moduli,
Springer Proc. Math. Stat. , Vol 409 (2023). Arxiv.
In this short note we show the unboundedness of the dimension of the K-moduli space of n-dimensional Fano varieties, and that the dimension of the stack can also be unbounded while, simultaneously, the dimension of the corresponding coarse space remains bounded.
P. Gallardo, J. Martinez-Garcia, C. Spotti, Applications of the moduli continuity method to log K-stable pairs.
J. Lond. Math. Soc. (2) 103 (2021), no. 2, 729–759. Arxiv.
The 'moduli continuity method' permits an explicit algebraisation of the Gromov-Hausdorff compactification of Kähler-Einstein metrics on Fano manifolds in some fundamental examples. In this paper, we apply such method in the 'log setting' to describe explicitly some compact moduli spaces of K-polystable log Fano pairs. We focus on situations when the angle of singularities is perturbed in an interval sufficiently close to one, by considering constructions arising from Geometric Invariant Theory. More precisely, we discuss the cases of pairs given by cubic surfaces with anticanonical sections, and of projective space with non-Fano hypersurfaces, and we show ampleness of the CM line bundle on their good moduli space (in the sense of Alper). Finally, we introduce a conjecture relating K-stability (and degenerations) of log pairs formed by a fixed Fano variety and pluri-anticanonical sections to certain natural GIT quotients.
M. de Borbon, C. Spotti, ALE Calabi-Yau metrics with conical singularities along a compact divisor,
IMRN, Issue 2, January 2021, Pages 1198–1223. Arxiv.
We construct ALE Calabi-Yau metrics with cone singularities along the exceptional set of resolutions of ℂn/Γ with non-positive discrepancies. In particular, this includes the case of the minimal resolution of two dimensional quotient singularities for any finite subgroup Γ⊂U(2) acting freely on the three-sphere, hence generalizing Kronheimer's construction of smooth ALE gravitational instantons. Finally, we show how our results extend to the more general asymptotically conical setting.
D. Angella, S. Calamai, C. Spotti, Remarks on Chern-Einstein Hermitian metrics.
Math. Z. 295 (2020), no. 3-4, 1707–1722. Arxiv.
We study some basic properties and examples of Hermitian metrics on complex manifolds whose traces of the curvature of the Chern connection are proportional to the metric itself.
M. de Borbon, C. Spotti, Local models for conical Kähler-Einstein metrics.
Proc. Amer. Math. Soc. 147 (2019), no. 3, 1217–1230. Arxiv.
In this note we use the Calabi ansatz, in the context of metrics with conical singularities along a divisor, to produce regular Calabi-Yau cones and Kähler-Einstein metrics of negative Ricci with a cuspidal point. As an application, we describe singularities and cuspidal ends of the completions of the complex hyperbolic metrics on the moduli spaces of ordered configurations of points in the projective line introduced by Thurston and Deligne-Mostow.
C. Spotti, S. Sun, Explicit Gromov-Hausdorff compactifications of moduli spaces of Kähler-Einstein Fano manifolds.
Pure Appl. Math. Q. (special volume for Donaldson 60) 13 (2017), no. 3, 477–515. Arxiv.
We exhibit the first non-trivial concrete examples of Gromov-Hausdorff compactifications of moduli spaces of Kähler-Einstein Fano manifolds in all complex dimensions bigger than two (Fano K-moduli spaces). We also discuss potential applications to explicit study of moduli spaces of K-stable Fano manifolds with large anti-canonical volume. Our arguments are based on recent progress about the geometry of metric tangent cones and on related ideas about the algebro-geometric study of singularities of K-stable Fano varieties.
C. Spotti, Kähler-Einstein metrics on Q-smoothable Fano varieties, their moduli and some applications,
Springer INdAM Proc., (2017). Arxiv.
We survey recent results on the existence of Kähler-Einstein metrics on certain smoothable Fano varieties, focusing on the importance of such metrics in the construction of compact algebraic moduli spaces of K-polystable Fano varieties. Moreover, we give some applications and we discuss some natural problems which deserve future investigations.
D. Angella, S. Calamai, C. Spotti, On the Chern-Yamabe problem.
Math. Res. Lett. 24 (2017), no. 3, 645–677. Arxiv.
We initiate the study of an analogue of the Yamabe problem for complex manifolds. More precisely, fixed a conformal Hermitian structure on a compact complex manifold, we are concerned in the existence of metrics with constant Chern scalar curvature. In this note, we set the problem and we provide a positive answer when the expected constant Chern scalar curvature is non-positive. In particular, this includes the case when the Kodaira dimension of the manifold is non-negative. Finally, we give some remarks on the positive curvature case, showing existence in some special cases and the failure, in general, of uniqueness of the solution.
C. Spotti, Cristiano, S. Sun, C. Yao Existence and deformations of Kähler-Einstein metrics on smoothable ℚ-Fano varieties.
Duke Math. J. 165 (2016), no. 16, 3043–3083. Arxiv.
We prove the existence of Kahler-Einstein metrics on Q-Gorenstein smoothable, K-polystable Q-Fano varieties, and we show how these metrics behave, in the Gromov-Hausdorff sense, under Q-Gorenstein smoothings.
C. Arezzo, C. Spotti, On cscK resolutions of conically singular cscK varieties.
J. Funct. Anal. 271 (2016), no. 2, 474–494. Arxiv.
In this note we discuss the problem of resolving conically singular cscK varieties to construct smooth cscK manifolds, showing a glueing result for (some) crepant resolutions of cscK varieties with discrete automorphism groups.
Y. Odaka, C. Spotti, S. Sun, Compact moduli spaces of del Pezzo surfaces and Kähler-Einstein metrics.
J. Diff. Geom. 102 (2016), no. 1, 127–172. Arxiv.
We prove that the Gromov-Hausdorff compactification of the moduli space of Kahler-Einstein Del Pezzo surfaces in each degree agrees with certain algebro-geometric compactification. In particular, this recovers Tian's theorem on the existence of Kahler-Einstein metrics on smooth Del Pezzo surfaces and classifies the degenerations of such metrics. The proof is based on a combination of both algebraic and differential geometric techniques.
C. Spotti, Deformations of nodal Kähler-Einstein del Pezzo surfaces with discrete automorphism groups.
J. Lond. Math. Soc. (2) 89 (2014), no. 2, 539–558. Arxiv.
In this paper we prove that generic small partial smoothings of Kahler-Einstein (KE) Del Pezzo orbifolds with only nodal singularities, and with no non-zero holomorphic vector fields, admit orbifold KE metrics which are close in the Gromov-Hausdorff sense to the original KE metric.
REPORTS
C. Spotti, Geometric aspects Kähler-Einstein metrics on klt pairs: remarks on optimal Bogomolov-Miyaoka-Yau inequality,
ZAG Handbook of Modern Algebraic Geometry (2025). Link.
C. Spotti, Towards canonical locally conformally Kähler metrics,
Oberwolfach report, Differentialgeometrie im Grossen (2023). Link.
C. Spotti, On geometric properties of log Kähler-Einstein metrics,
Oberwolfach report, Differentialgeometrie im Grossen (2019). Link
LECTURE NOTES & SURVEYS
D. Angella, C. Spotti, Kähler-Einstein metrics old and new,
Complex Geometry, 4 (2017). Arxiv.
We present classical and recent results on Kähler-Einstein metrics on compact complex manifolds, focusing on existence, obstructions and relations to algebraic geometric notions of stability (K-stability). These are the notes for the SMI course "Kähler-Einstein metrics" given by C.S. in Cortona (Italy), May 2017. The material is not intended to be original.
C. Spotti, Kähler-Einstein metrics via moduli continuity method,
Springer INdAM Proc., (2017). Link.
We discuss some ideas behind a strategy that has been used to construct Kähler-Einstein metrics for explicit families of Fano varieties.
PHD Thesis
C. Spotti, Degenerations of Kähler-Einstein Fano manifolds, PhD Thesis Imperial College London,
available on arXiv:1211.533 (2012).