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1. Variation of singular Kähler–Einstein metrics: Positive Kodaira dimension

   https://doi.org/10.1515/crelle-2021-0028  

   Cao, Junyan; Guenancia, Henri; Păun, Mihai (August 2021, Journal für die reine und angewandte Mathematik (Crelles Journal))

   null (Ed.)

   Abstract Given a Kähler fiber space p : X → Y {p:X\to Y} whose generic fiber is of general type, we prove that the fiberwise singular Kähler–Einstein metric inducesa semipositively curved metric on the relative canonical bundle K X / Y {K_{X/Y}} of p . We also propose a conjectural generalization of this result for relativetwisted Kähler–Einstein metrics. Then we show that our conjecture holds trueif the Lelong numbers of the twisting current are zero. Finally, we explain the relevance of our conjecture for the study of fiberwise Song–Tian metrics (which represent the analogue of KE metrics for fiber spaces whose generic fiber has positive but not necessarily maximal Kodaira dimension). 

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2. A generalization of Geroch's conjecture

   https://doi.org/10.1002/cpa.22137  

   Brendle, Simon; Hirsch, Sven; Johne, Florian (January 2024, Communications on Pure and Applied Mathematics)

   Abstract The Theorem of Bonnet–Myers implies that manifolds with topology do not admit a metric of positive Ricci curvature, while the resolution of Geroch's conjecture implies that the torus does not admit a metric of positive scalar curvature. In this work we introduce a new notion of curvature interpolating between Ricci and scalar curvature (so‐calledm‐intermediate curvature), and use stable weighted slicings to show that for and the manifolds do not admit a metric of positivem‐intermediate curvature. 

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3. On a generalized conjecture of Hopf with symmetry

   https://doi.org/10.1112/S0010437X16008150  

   Amann, Manuel; Kennard, Lee (February 2017, Compositio Mathematica)

   A famous conjecture of Hopf states that $$\mathbb{S}^{2}\times \mathbb{S}^{2}$$ does not admit a Riemannian metric with positive sectional curvature. In this article, we prove that no manifold product $$N\times N$$ can carry a metric of positive sectional curvature admitting a certain degree of torus symmetry. 

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4. Bergman–Einstein metrics, a generalization of Kerner’s theorem and Stein spaces with spherical boundaries

   https://doi.org/10.1515/crelle-2020-0012  

   Huang, Xiaojun; Xiao, Ming (January 2021, Journal für die reine und angewandte Mathematik (Crelles Journal))

   null (Ed.)

   Abstract We give an affirmative solution to a conjecture of Cheng proposed in 1979which asserts that the Bergman metric of a smoothly bounded stronglypseudoconvex domain in {\mathbb{C}^{n},n\geq 2} , is Kähler–Einsteinif and only if the domain is biholomorphic to the ball. We establisha version of the classical Kerner theorem for Stein spaces withisolated singularities which has an immediate application toconstruct a hyperbolic metric over a Stein space with a sphericalboundary. 

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5. A relative version of the Turaev–Viro invariants and the volume of hyperbolic polyhedral 3‐manifolds

   https://doi.org/10.1112/topo.12300  

   Yang, Tian (May 2023, Journal of Topology)

   Abstract We define a relative version of the Turaev–Viro invariants for an ideally triangulated compact 3‐manifold with nonempty boundary and a coloring on the edges, generalizing the Turaev–Viro invariants [36] of the manifold. We also propose the volume conjecture for these invariants whose asymptotic behavior is related to the volume of the manifold in the hyperbolic polyhedral metric [22, 23] with singular locus of the edges and cone angles determined by the coloring, and prove the conjecture in the case that the cone angles are sufficiently small. This suggests an approach of solving the volume conjecture for the Turaev–Viro invariants proposed by Chen–Yang [8] for hyperbolic 3‐manifolds with totally geodesic boundary. 

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