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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. Six‐flows on almost balanced signed graphs

   https://doi.org/10.1002/jgt.22460  

   Wang, Xiao; Lu, You; Zhang, Cun‐Quan; Zhang, Shenggui (April 2019, Journal of Graph Theory)

   Abstract In 1983, Bouchet conjectured that every flow‐admissible signed graph admits a nowhere‐zero 6‐flow. By Seymour's 6‐flow theorem, Bouchet's conjecture holds for signed graphs with all edges positive. Recently, Rollová et al proved that every flow‐admissible signed cubic graph with two negative edges admits a nowhere‐zero 7‐flow, and admits a nowhere‐zero 6‐flow if its underlying graph either contains a bridge, or is 3‐edge‐colorable, or is critical. In this paper, we improve and extend these results, and confirm Bouchet's conjecture for signed graphs with frustration number at most two, where the frustration number of a signed graph is the smallest number of vertices whose deletion leaves a balanced signed graph. 

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3. Generalized Alder-Type Partition Inequalities

   https://doi.org/10.37236/11606  

   Armstrong, Liam; Ducasse, Bryan; Meyer, Thomas; Swisher, Holly (June 2023, The Electronic Journal of Combinatorics)

   In 2020, Kang and Park conjectured a "level $$2$$" Alder-type partition inequality which encompasses the second Rogers-Ramanujan Identity. Duncan, Khunger, the fourth author, and Tamura proved Kang and Park's conjecture for all but finitely many cases utilizing a "shift" inequality and conjectured a further, weaker generalization that would extend both Alder's (now proven) as well as Kang and Park's conjecture to general level. Utilizing a modified shift inequality, Inagaki and Tamura have recently proven that the Kang and Park conjecture holds for level $$3$$ in all but finitely many cases. They further conjectured a stronger shift inequality which would imply a general level result for all but finitely many cases. Here, we prove their conjecture for large enough $$n$$, generalize the result for an arbitrary shift, and discuss the implications for Alder-type partition inequalities. 

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4. Asymptotics of the Poisson kernel and Green's functions of the fractional conformal Laplacian

   https://doi.org/10.3934/dcds.2022085  

   Mayer, Martin; Ndiaye, Cheikh Birahim (January 2022, Discrete and Continuous Dynamical Systems)

   We study the asymptotics of the Poisson kernel and Green's functions of the fractional conformal Laplacian for conformal infinities of asymptotically hyperbolic manifolds. We derive sharp expansions of the Poisson kernel and Green's functions of the conformal Laplacian near their singularities. Our expansions of the Green's functions answer the first part of the conjecture of Kim-Musso-Wei[21] in the case of locally flat conformal infinities of Poincare-Einstein manifolds and together with the Poisson kernel asymptotic is used also in our paper [25] to show solvability of the fractional Yamabe problem in that case. Our asymptotics of the Green's functions on the general case of conformal infinities of asymptotically hyperbolic space is used also in [29] to show solvability of the fractional Yamabe problem for conformal infinities of dimension \begin{document}$ 3 $$\end{document} and fractional parameter in \begin{document}$$ (\frac{1}{2}, 1) $$\end{document}$ corresponding to a global case left by previous works. 

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5. THE INTEGRAL HODGE CONJECTURE FOR 3-FOLDS OF KODAIRA DIMENSION ZERO

   https://doi.org/10.1017/S1474748019000665  

   Totaro, Burt (September 2021, Journal of the Institute of Mathematics of Jussieu)

   Abstract We prove the integral Hodge conjecture for all 3-folds $$X$$ of Kodaira dimension zero with $$H^{0}(X,K_{X})$$ not zero. This generalizes earlier results of Voisin and Grabowski. The assumption is sharp, in view of counterexamples by Benoist and Ottem. We also prove similar results on the integral Tate conjecture. For example, the integral Tate conjecture holds for abelian 3-folds in any characteristic. 

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