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1. Jacobian determinants for nonlinear gradient of planar ∞-harmonic functions and applications

   https://doi.org/10.1515/crelle-2024-0016  

   Dong, Hongjie; Peng, Fa; Zhang, Yi Ru-Ya; Zhou, Yuan (April 2024, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We introduce a distributional Jacobian determinant $detD{V}_{\beta }\left(Dv\right)$ \det DV_{\beta}(Dv)in dimension two for the nonlinear complex gradient $V_{\beta }\left(Dv\right)={\left|Dv\right|}^{\beta }(v_{x_1},-v_{x_2})$ V_{\beta}(Dv)=\lvert Dv\rvert^{\beta}(v_{x_{1}},-v_{x_{2}})for any $\beta >-1$ \beta>-1, whenever $v\in W_{\mathrm{loc}}^{1,2}$ v\in W^{1\smash{,}2}_{\mathrm{loc}}and $\beta {\left|Dv\right|}^{1+\beta }\in W_{\mathrm{loc}}^{1,2}$ \beta\lvert Dv\rvert^{1+\beta}\in W^{1\smash{,}2}_{\mathrm{loc}}.This is new when $\beta \ne 0$ \beta\neq 0.Given any planar ∞-harmonic function 𝑢, we show that such distributional Jacobian determinant $detD{V}_{\beta }\left(Du\right)$ \det DV_{\beta}(Du)is a nonnegative Radon measure with some quantitative local lower and upper bounds.We also give the following two applications. Applying this result with $\beta =0$ \beta=0, we develop an approach to build up a Liouville theorem, which improves that of Savin.Precisely, if 𝑢 is an ∞-harmonic function in the whole ${\mathbb{R}}^2$ \mathbb{R}^{2}with $\underset{R\to \mathrm{\infty }}{lim\; inf}\underset{c\in \mathbb{R}}{inf}\frac{1}{R}{⨍}_{B(0,R)}\left|u\left(x\right)-c\right|dx<\mathrm{\infty },$ \liminf_{R\to\infty}\inf_{c\in\mathbb{R}}\frac{1}{R}\barint_{B(0,R)}\lvert u(x)-c\rvert\,dx<\infty,then $u=b+a\cdot x$ u=b+a\cdot xfor some $b\in \mathbb{R}$ b\in\mathbb{R}and $a\in {\mathbb{R}}^2$ a\in\mathbb{R}^{2}.Denoting by $u_p$ u_{p}the 𝑝-harmonic function having the same nonconstant boundary condition as 𝑢, we show that $detD{V}_{\beta }\left(D{u}_p\right)\to detD{V}_{\beta }\left(Du\right)$ \det DV_{\beta}(Du_{p})\to\det DV_{\beta}(Du)as $p\to \mathrm{\infty }$ p\to\inftyin the weak-⋆ sense in the space of Radon measure.Recall that $V_{\beta }\left(D{u}_p\right)$ V_{\beta}(Du_{p})is always quasiregular mappings, but $V_{\beta }\left(Du\right)$ V_{\beta}(Du)is not in general. 

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2. K-moduli of Fano threefolds and genus four curves

   https://doi.org/10.1515/crelle-2025-0016  

   Liu, Yuchen; Zhao, Junyan (March 2025, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract In this article, we study the K-moduli space of Fano threefolds obtained by blowing up ${\mathbb{P}}^3$ \mathbb{P}^{3}along $(2,3)$ (2,3)-complete intersection curves.This K-moduli space is a two-step birational modification of the GIT moduli space of bidegree $(3,3)$ (3,3)-curves on ${\mathbb{P}}^1\times {\mathbb{P}}^1$ \mathbb{P}^{1}\times\mathbb{P}^{1}.As an application, we show that our K-moduli space appears as one model of the Hassett–Keel program for ${\bar{M}}_4$ \overline{M}_{4}.In particular, we classify all K-(semi/poly)stable members in this deformation family of Fano varieties.We follow the moduli continuity method with moduli of lattice-polarized K3 surfaces, general elephants and Sarkisov links as new ingredients. 

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3. A new zero-free region for Rankin–Selberg 𝐿-functions

   https://doi.org/10.1515/crelle-2025-0009  

   Harcos, Gergely; Thorner, Jesse (March 2025, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Let 𝜋 and ${\pi }^{\prime }$ \pi^{\prime}be cuspidal automorphic representations of $\mathrm{GL}\left(n\right)$ \mathrm{GL}(n)and $\mathrm{GL}\left(n^{\prime }\right)$ \mathrm{GL}(n^{\prime})with unitary central characters.We establish a new zero-free region for all $\mathrm{GL}\left(1\right)$ \mathrm{GL}(1)-twists of the Rankin–Selberg 𝐿-function $L(s,\pi \times {\pi }^{\prime })$ L(s,\pi\times\pi^{\prime}), generalizing Siegel’s celebrated work on Dirichlet 𝐿-functions.As an application, we prove the first unconditional Siegel–Walfisz theorem for the Dirichlet coefficients of $-L^{\prime }(s,\pi \times {\pi }^{\prime })/L(s,\pi \times {\pi }^{\prime })$ -L^{\prime}(s,\pi\times\pi^{\prime})/L(s,\pi\times\pi^{\prime}).Also, for $n\le 8$ n\leq 8, we extend the region of holomorphy and nonvanishing for the twisted symmetric power 𝐿-functions $L(s,\pi ,{\mathrm{Sym}}^n\otimes \chi )$ L(s,\pi,\mathrm{Sym}^{n}\otimes\chi)of any cuspidal automorphic representation of $\mathrm{GL}\left(2\right)$ \mathrm{GL}(2). 

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4. Higher order Kirillov--Reshetikhin modules for 𝐔 _q ( A _n ⁽¹⁾ ), imaginary modules and monoidal categorification

   https://doi.org/10.1515/crelle-2023-0068  

   Brito, Matheus; Chari, Vyjayanthi (October 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We study the family of irreducible modules for quantum affine $𝔰{𝔩}_{n+1}$ {\mathfrak{sl}_{n+1}}whose Drinfeld polynomials are supported on just one node of the Dynkin diagram. We identify all the prime modules in this family and prove a unique factorization theorem. The Drinfeld polynomials of the prime modules encode information coming from the points of reducibility of tensor products of the fundamental modules associated to $A_m$ {A_{m}}with $m\le n$ {m\leq n}. These prime modules are a special class of the snake modules studied by Mukhin and Young. We relate our modules to the work of Hernandez and Leclerc and define generalizations of the category ${\mathcal{𝒞}}^{-}$ {\mathscr{C}^{-}}. This leads naturally to the notion of an inflation of the corresponding Grothendieck ring. In the last section we show that the tensor product of a (higher order) Kirillov–Reshetikhin module with its dual always contains an imaginary module in its Jordan–Hölder series and give an explicit formula for its Drinfeld polynomial. Together with the results of [D. Hernandez and B. Leclerc,A cluster algebra approach toq-characters of Kirillov–Reshetikhin modules,J. Eur. Math. Soc. (JEMS) 18 2016, 5, 1113–1159] this gives examples of a product of cluster variables which are not in the span of cluster monomials. We also discuss the connection of our work with the examples arising from the work of [E. Lapid and A. Mínguez,Geometric conditions for $\mathrm{\square }$ \square-irreducibility of certain representations of the general linear group over a non-archimedean local field,Adv. Math. 339 2018, 113–190]. Finally, we use our methods to give a family of imaginary modules in type $D_4$ {D_{4}}which do not arise from an embedding of $A_r$ {A_{r}}with $r\le 3$ {r\leq 3}in $D_4$ {D_{4}}. 

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5. Ordinary modules for vertex algebras of 𝔬𝔰𝔭 _1|2𝑛

   https://doi.org/10.1515/crelle-2024-0060  

   Creutzig, Thomas; Genra, Naoki; Linshaw, Andrew (August 2024, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We show that the affine vertex superalgebra $V^k\left(\mathfrak{o}\mathfrak{s}{\mathfrak{p}}_{1|2n}\right)$ V^{k}(\mathfrak{osp}_{1|2n})at generic level 𝑘 embeds in the equivariant 𝒲-algebra of $\mathfrak{s}{\mathfrak{p}}_{2n}$ \mathfrak{sp}_{2n}times $4n$ 4nfree fermions.This has two corollaries:(1) it provides a new proof that, for generic 𝑘, the coset $\mathrm{Com}(V^k\left(\mathfrak{s}{\mathfrak{p}}_{2n}\right),V^k\left(\mathfrak{o}\mathfrak{s}{\mathfrak{p}}_{1|2n}\right))$ \operatorname{Com}(V^{k}(\mathfrak{sp}_{2n}),V^{k}(\mathfrak{osp}_{1|2n}))is isomorphic to ${\mathcal{W}}^{\mathrm{\ell }}\left(\mathfrak{s}{\mathfrak{p}}_{2n}\right)$ \mathcal{W}^{\ell}(\mathfrak{sp}_{2n})for $\mathrm{\ell }=-\left(n+1\right)+\left(k+n+1\right)/\left(2k+2n+1\right)$ \ell=-(n+1)+(k+n+1)/(2k+2n+1), and(2) we obtain the decomposition of ordinary $V^k\left(\mathfrak{o}\mathfrak{s}{\mathfrak{p}}_{1|2n}\right)$ V^{k}(\mathfrak{osp}_{1|2n})-modules into $V^k\left(\mathfrak{s}{\mathfrak{p}}_{2n}\right)\otimes {\mathcal{W}}^{\mathrm{\ell }}\left(\mathfrak{s}{\mathfrak{p}}_{2n}\right)$ V^{k}(\mathfrak{sp}_{2n})\otimes\mathcal{W}^{\ell}(\mathfrak{sp}_{2n})-modules.Next, if 𝑘 is an admissible level and ℓ is a non-degenerate admissible level for $\mathfrak{s}{\mathfrak{p}}_{2n}$ \mathfrak{sp}_{2n}, we show that the simple algebra $L_k\left(\mathfrak{o}\mathfrak{s}{\mathfrak{p}}_{1|2n}\right)$ L_{k}(\mathfrak{osp}_{1|2n})is an extension of the simple subalgebra $L_k\left(\mathfrak{s}{\mathfrak{p}}_{2n}\right)\otimes {\mathcal{W}}_{\mathrm{\ell }}\left(\mathfrak{s}{\mathfrak{p}}_{2n}\right)$ L_{k}(\mathfrak{sp}_{2n})\otimes{\mathcal{W}}_{\ell}(\mathfrak{sp}_{2n}).Using the theory of vertex superalgebra extensions, we prove that the category of ordinary $L_k\left(\mathfrak{o}\mathfrak{s}{\mathfrak{p}}_{1|2n}\right)$ L_{k}(\mathfrak{osp}_{1|2n})-modules is a semisimple, rigid vertex tensor supercategory with only finitely many inequivalent simple objects.It is equivalent to a certain subcategory of ${\mathcal{W}}_{\mathrm{\ell }}\left(\mathfrak{s}{\mathfrak{p}}_{2n}\right)$ \mathcal{W}_{\ell}(\mathfrak{sp}_{2n})-modules.A similar result also holds for the category of Ramond twisted modules.Due to a recent theorem of Robert McRae, we get as a corollary that categories of ordinary $L_k\left(\mathfrak{s}{\mathfrak{p}}_{2n}\right)$ L_{k}(\mathfrak{sp}_{2n})-modules are rigid. 

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