https://neckpinch.com/authors/andrea-loi/New arXiv papers by Andrea Loi, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2510.06405arxiv:2510.06405Tue, 07 Oct 2025 00:00:00 GMTAndrea Loi, Roberto Mossa, Fabio ZUddas — We study the Bergman metric and introduce the Bergman dual on Cartan-Hartogs (CH) domains. For a bounded domain D in C^n with Bergman kernel K_D, we define the Bergman dual of (D, g_D) as (D*, g_D*), where D* is the maximal domain on which the modified kernel K_D*(z, zbar) = K_D(z, -zbar) is positive, and g_D* is the Kahler metric obtained from K_D*. For a Cartan-Hartogs domain M_Omega, mu we prove the equivalence of: (i) M_Omega, mu is biholomorphic to the unit ball; (ii) its Bergman metric is a Kahler-Ricci soliton; (iii) after rescaling by a constant factor, the Bergman dual is finitely projectively induced. Conditions (i) and (ii) are Bergman-metric analogues of classical rigidity for Kahler-Einstein metrics (related to Yau's problem and Cheng's conjecture) and to recent rigidity for Kahler-Ricci solitons. Condition (iii) emphasizes the duality viewpoint, inspired by bounded symmetric domains and their compact duals. We also compare our results with other canonical metrics on CH domains, namely g_Omega, mu and hat g_Omega, mu, and discuss open problems about the maximal domain on which the Bergman dual is defined.Ricci solitonsKähler–Ricci solitonsEinstein metricsKähler geometryStability & uniquenessKähler–Einstein & K-stability https://arxiv.org/abs/2507.23606arxiv:2507.23606Thu, 31 Jul 2025 00:00:00 GMTAndrea Loi, Roberto Mossa, Fabio Zuddas — We prove a universal embedding theorem for flag manifolds: every flag manifold admits a holomorphic isometric embedding into an irreducible classical flag manifold. This result generalizes the classical celebrated embedding theorems of Takeuchi [30] and Nakagawa-Takagi [27]. Using this embedding, we establish new rigidity phenomena for holomorphic isometries between homogeneous Kähler manifolds. As a first immediate consequence we show the triviality of a Kähler-Ricci soliton submanifod of C × Ω, where C is a flag manifold and Ω is a homogeneous bounded domain. Secondly, we show that no weak-relative relationship can occur among the fundamental classes of homogeneous Kähler manifolds: flat spaces, flag manifolds, and homogeneous bounded domains. Two Kähler manifolds are said to be weak relatives if they share, up to local isometry, a common Kähler submanifold of complex dimension at least two. Our main result precisely shows that if E is (possibly indefinite) flat, C is a flag manifold, and Ω is a homogeneous bounded domain, then: E is not weak relative to C×Ω; C is not weak relative to E×Ω; Ω is not weak relative to E× C. This extends, in two independent directions, the rigidity theorem of Loi-Mossa [22]: we pass from relatives to the more flexible notion of weak relatives and dispense with the earlier "special" restriction on the flag-manifold factor. This result also unifies previous rigidity results from the literature, e.g., [5, 6, 7, 9, 12, 13, 32].Ricci solitonsKähler–Ricci solitonsKähler geometryHomogeneous spacesStability & uniqueness https://arxiv.org/abs/2307.11500arxiv:2307.11500Fri, 21 Jul 2023 00:00:00 GMTAndrea Loi, Giovanni Placini — We introduce a large class of canonical Kähler metrics, called in this paper well-behaved, extending metrics induced by complex space forms. We study Kähler–Ricci iterations of well-behaved metrics on compact and non-compact Kähler manifolds. That is, we are interested in well-behaved metrics for which the iteration of the Ricci operator is a multiple of a Kähler metric, i.e., ρ_ω^k=λΩ. In particular, when k=1, under some condition on the maximal domain of definition of canonical coordinates, we show that λ is forced to be positive. Moreover, for arbitrary k, we prove two additional results. Namely, if ω and Ω are induced by a flat metric, then ω is Ricci-flat. Finally, if a Kähler-Ricci soliton Ω arises as Kähler–Ricci iteration of a metric ω induced by a complex space form, then the Kähler–Ricci soliton is forced to be trivial, that is, Kähler–Einstein. These three theorems extend well known results on Kähler–Einstein metrics to higher iterations of the Ricci operator and a larger class of metrics.Ricci solitonsKähler–Ricci solitonsEinstein metricsRicci-flat & Calabi–YauSphere theoremsKähler geometryBoundaries & non-compactKähler–Einstein & K-stability https://arxiv.org/abs/2305.05244arxiv:2305.05244Tue, 09 May 2023 00:00:00 GMTA. Loi, R. Mossa — We prove two rigidity results on holomorphic isometries into homogeneous Kähler manifolds. The first shows that a Kähler-Ricci soliton induced by the homogeneous metric of the Kähler product of a special flag manifold (i.e. a flag of classical type or integral type) with a bounded homogeneous domain is trivial, i.e. Kähler-Einstein. In the second one we prove that: (i) a flat space is not relative to the Kähler product of a special flag manifold with a homogeneous bounded domain, (ii) a special flag manifold is not relative to the Kähler product of a flat space with a homogeneous bounded domain and (iii) a homogeneous bounded domain is not relative to the Kähler product of a flat space with a special flag manifold. Our theorems strongly extend the results in [4], [5], [12], [13] and [22].Ricci solitonsKähler–Ricci solitonsEinstein metricsKähler geometryHomogeneous spacesStability & uniquenessKähler–Einstein & K-stability https://arxiv.org/abs/2205.11297arxiv:2205.11297Mon, 23 May 2022 00:00:00 GMTAndrea Loi, Roberto Mossa — We prove two rigidity theorems on holomorphic isometries into homogeneous bounded domains. The first shows that a Kähler-Ricci soliton induced by the homogeneous metric of a homogeneous bounded domain is trivial, i.e. Kähler-Einstein. In the second one we prove that a homogeneous bounded domain and the flat (definite or indefinite) complex Euclidean space are not relatives, i.e. they do not share a common Kähler submanifold (of positive dimension). Our theorems extend the results proved in [A. Loi, R. Mossa, Proc. Amer. Math. Soc. 149 (2021), no. 11, 4931-4941] and [X. Cheng, Y. Hao, Ann. Global Anal. Geom. 60 (2021), no. 1, 167-180] respectively.Ricci solitonsKähler–Ricci solitonsEinstein metricsKähler geometryHomogeneous spacesStability & uniquenessKähler–Einstein & K-stability https://arxiv.org/abs/2109.15229arxiv:2109.15229Thu, 30 Sep 2021 00:00:00 GMTAndrea Loi, Filippo Salis, Fabio Zuddas — We prove that a radial Kaehler metric g is Kaehler-Einstein if and only if one of the following conditions is satisfied: 1. g is extremal and it is associated to a Kaehler-Ricci soliton; 2. two different generalized scalar curvatures of g are constant; 3. g is extremal (not cscK) and one of its generalized scalar curvature is constant.Ricci solitonsKähler–Ricci solitonsEinstein metricsKähler geometryScalar curvatureKähler–Einstein & K-stability