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1. On finite time Type I singularities of the Kähler–Ricci flow on compact Kähler surfaces

   https://doi.org/10.4171/JEMS/1485  

   Cifarelli, Charles; Conlon, Ronan J.; Deruelle, Alix (July 2024, Journal of the European Mathematical Society)

   We show that the underlying complex manifold of a complete non-compact two-dimensional shrinking gradient Kähler-Ricci soliton (M,g,X) with soliton metric g with bounded scalar curvature Rg whose soliton vector field X has an integral curve along which Rg↛0 is biholomorphic to either C×P1 or to the blowup of this manifold at one point. Assuming the existence of such a soliton on this latter manifold, we show that it is toric and unique. We also identify the corresponding soliton vector field. Given these possibilities, we then prove a strong form of the Feldman-Ilmanen-Knopf conjecture for finite time Type I singularities of the Kähler-Ricci flow on compact Kähler surfaces, leading to a classification of the bubbles of such singularities in this dimension. 

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2. Moduli space reconstruction and Weak Gravity

   https://doi.org/10.1007/JHEP12(2023)134  

   Gendler, Naomi; Heidenreich, Ben; McAllister, Liam; Moritz, Jakob; Rudelius, Tom (December 2023, Journal of High Energy Physics)

   We present a method to construct the extended Kähler cone of any Calabi-Yau threefold by using Gopakumar-Vafa invariants to identify all geometric phases that are related by flops or Weyl reflections. In this way we obtain the Kähler moduli spaces of all favorable Calabi-Yau threefold hypersurfaces with h1,1 ≤ 4, including toric and non-toric phases. In this setting we perform an explicit test of the Weak Gravity Conjecture by using the Gopakumar-Vafa invariants to count BPS states. All of our examples satisfy the tower/sublattice WGC, and in fact they even satisfy the stronger lattice WGC. 

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3. The Riemannian and symplectic geometry of the space of generalized Kähler structures

   https://doi.org/10.4171/CMH/585  

   Apostolov, Vestislav; Streets, Jeffrey; Ustinovskiy, Yury (February 2025, Commentarii Mathematici Helvetici)

   On a compact complex manifold(M, J)endowed with a holomorphic Poisson tensor \pi_{J}and a de Rham class\alpha\in H^{2}(M, \mathbb{R}), we study the space of generalized Kähler (GK) structures defined by a symplectic formF\in \alphaand whose holomorphic Poisson tensor is\pi_{J}. We define a notion of generalized Kähler class of such structures, and use the moment map framework of Boulanger (2019) and Goto (2020) to extend the Calabi program to GK geometry. We obtain generalizations of the Futaki–Mabuchi extremal vector field (1995) and the Calabi–Lichnerowicz–Matsushima result (1982, 1958, 1957) for the Lie algebra of the group of automorphisms of(M, J, \pi_{J}). We define a closed1-form on a GK class, which yields a generalization of the Mabuchi energy and thus a variational characterization of GK structures of constant scalar curvature. Next we introduce a formal Riemannian metric on a given GK class, generalizing the fundamental construction of Mabuchi–Semmes–Donaldson (1987, 1992, 1997) We show that this metric has nonpositive sectional curvature, and that the Mabuchi energy is convex along geodesics, leading to a conditional uniqueness result for constant scalar curvature GK structures. We finally examine the toric case, proving the uniqueness of extremal generalized Kähler structures and showing that their existence is obstructed by the uniform relative K-stability of the corresponding Delzant polytope. Using the resolution of the Yau–Tian–Donaldson conjecture in the toric case by Chen–Cheng (2021) and He (2019), we show in some settings that this condition suffices for existence and thus construct new examples. 

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4. Gravitational Instantons, Weyl Curvature, and Conformally Kähler Geometry

   https://doi.org/10.1093/imrn/rnae200  

   Biquard, Olivier; Gauduchon, Paul; LeBrun, Claude (September 2024, International Mathematics Research Notices)

   Abstract In a previous paper [7], the first two authors classified complete Ricci-flat ALF Riemannian 4-manifolds that are toric and Hermitian, but non-Kähler. In this article, we consider general Ricci-flat metrics on these spaces that are close to a given such gravitational instanton with respect to a norm that imposes reasonable fall-off conditions at infinity. We prove that any such Ricci-flat perturbation is necessarily Hermitian and carries a bounded, non-trivial Killing vector field. With mild additional hypotheses, we are then able to show that the new Ricci-flat metric must actually be one of the known gravitational instantons classified in [7]. 

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5. A variational approach to the Yau–Tian–Donaldson conjecture

   https://doi.org/10.1090/jams/964  

   Berman, Robert; Boucksom, Sébastien; Jonsson, Mattias (July 2021, Journal of the American Mathematical Society)

   We give a variational proof of a version of the Yau–Tian–Donaldson conjecture for twisted Kähler–Einstein currents, and use this to express the greatest (twisted) Ricci lower bound in terms of a purely algebro-geometric stability threshold. Our approach does not involve a continuity method or the Cheeger–Colding–Tian theory, and uses instead pluripotential theory and valuations. Along the way, we study the relationship between geodesic rays and non-Archimedean metrics. 

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