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1. Inequalities for the Radon Transform on Convex Sets

   https://doi.org/10.1093/imrn/rnab122  

   Giannopoulos, Apostolos; Koldobsky, Alexander; Zvavitch, Artem (May 2021, International Mathematics Research Notices)

   Abstract We prove an inequality that unifies previous works of the authors on the properties of the Radon transform on convex bodies including an extension of the Busemann–Petty problem and a slicing inequality for arbitrary functions. Let $$K$$ and $$L$$ be star bodies in $${\mathbb R}^n,$$ let $0<k<n$ be an integer, and let $f,g$ be non-negative continuous functions on $$K$$ and $$L$$, respectively, so that $$\|g\|_\infty =g(0)=1.$$ Then $$\begin{align*} & \frac{\int_Kf}{\left(\int_L g\right)^{\frac{n-k}n}|K|^{\frac kn}} \le \frac n{n-k} \left(d_{\textrm{ovr}}(K,\mathcal{B}\mathcal{P}_k^n)\right)^k \max_{H} \frac{\int_{K\cap H} f}{\int_{L\cap H} g}, \end{align*}$$where $|K|$ stands for volume of proper dimension, $$C$$ is an absolute constant, the maximum is taken over all $(n-k)$-dimensional subspaces of $${\mathbb R}^n,$$ and $$d_{\textrm{ovr}}(K,\mathcal{B}\mathcal{P}_k^n)$$ is the outer volume ratio distance from $$K$$ to the class of generalized $$k$$-intersection bodies in $${\mathbb R}^n.$$ Another consequence of this result is a mean value inequality for the Radon transform. We also obtain a generalization of the isomorphic version of the Shephard problem. 

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2. The Cohomological Brauer Group of a Torsion $${\mathbb{G}}_{m}$$-Gerbe

   https://doi.org/10.1093/imrn/rnz235  

   Shin, Minseon (November 2019, International Mathematics Research Notices)

   Abstract Let $$S$$ be a scheme and let $$\pi : \mathcal{G} \to S$$ be a $${\mathbb{G}}_{m,S}$$-gerbe corresponding to a torsion class $$[\mathcal{G}]$$ in the cohomological Brauer group $${\operatorname{Br}}^{\prime}(S)$$ of $$S$$. We show that the cohomological Brauer group $${\operatorname{Br}}^{\prime}(\mathcal{G})$$ of $$\mathcal{G}$$ is isomorphic to the quotient of $${\operatorname{Br}}^{\prime}(S)$$ by the subgroup generated by the class $$[\mathcal{G}]$$. This is analogous to a theorem proved by Gabber for Brauer–Severi schemes. 

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3. Maps from Feigin and Odesskii's elliptic algebras to twisted homogeneous coordinate rings

   https://doi.org/10.1017/fms.2020.60  

   Chirvasitu, Alex; Kanda, Ryo; Smith, S. Paul (January 2021, Forum of Mathematics, Sigma)

   null (Ed.)

   Abstract The elliptic algebras in the title are connected graded $$\mathbb {C}$$ -algebras, denoted $$Q_{n,k}(E,\tau )$$ , depending on a pair of relatively prime integers $$n>k\ge 1$$ , an elliptic curve E and a point $$\tau \in E$$ . This paper examines a canonical homomorphism from $$Q_{n,k}(E,\tau )$$ to the twisted homogeneous coordinate ring $$B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$$ on the characteristic variety $$X_{n/k}$$ for $$Q_{n,k}(E,\tau )$$ . When $$X_{n/k}$$ is isomorphic to $E^g$ or the symmetric power $S^gE$ , we show that the homomorphism $$Q_{n,k}(E,\tau ) \to B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$$ is surjective, the relations for $$B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$$ are generated in degrees $$\le 3$$ and the noncommutative scheme $$\mathrm {Proj}_{nc}(Q_{n,k}(E,\tau ))$$ has a closed subvariety that is isomorphic to $E^g$ or $S^gE$ , respectively. When $$X_{n/k}=E^g$$ and $$\tau =0$$ , the results about $$B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$$ show that the morphism $$\Phi _{|\mathcal {L}_{n/k}|}:E^g \to \mathbb {P}^{n-1}$$ embeds $E^g$ as a projectively normal subvariety that is a scheme-theoretic intersection of quadric and cubic hypersurfaces. 

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4. Differentiating Siegel Modular Forms and the Moving Slope of 𝒜 g

   https://doi.org/10.1093/imrn/rnad203  

   Grushevsky, Samuel; Ibukiyama, Tomoyoshi; Mondello, Gabriele; Salvati_Manni, Riccardo (August 2023, International Mathematics Research Notices)

   Abstract We study the cone of moving divisors on the moduli space $${\mathcal{A}}_{g}$$ of principally polarized abelian varieties. Partly motivated by the generalized Rankin–Cohen bracket, we construct a non-linear holomorphic differential operator that sends Siegel modular forms to Siegel modular forms, and we apply it to produce new modular forms. Our construction recovers the known divisors of minimal moving slope on $${\mathcal{A}}_{g}$$ for $$g\leq 4$$, and gives an explicit upper bound for the moving slope of $${\mathcal{A}}_{5}$$ and a conjectural upper bound for the moving slope of $${\mathcal{A}}_{6}$$. 

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5. Almost isotropic Kähler manifolds

   https://doi.org/10.1515/crelle-2019-0030  

   Schmidt, Benjamin; Shankar, Krishnan; Spatzier, Ralf (October 2020, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Let M be a complete Riemannian manifold and suppose {p\in M} . For each unit vector {v\in T_{p}M} , the Jacobi operator , {\mathcal{J}_{v}:v^{\perp}\rightarrow v^{\perp}} is the symmetric endomorphism, {\mathcal{J}_{v}(w)=R(w,v)v} . Then p is an isotropic point if there exists a constant {\kappa_{p}\in{\mathbb{R}}} such that {\mathcal{J}_{v}=\kappa_{p}\operatorname{Id}_{v^{\perp}}} for each unit vector {v\in T_{p}M} . If all points are isotropic, then M is said to be isotropic; it is a classical result of Schur that isotropic manifolds of dimension at least 3 have constant sectional curvatures. In this paper we consider almost isotropic manifolds , i.e. manifolds having the property that for each {p\in M} , there exists a constant {\kappa_{p}\in\mathbb{R}} such that the Jacobi operators {\mathcal{J}_{v}} satisfy {\operatorname{rank}({\mathcal{J}_{v}-\kappa_{p}\operatorname{Id}_{v^{\perp}}}% )\leq 1} for each unit vector {v\in T_{p}M} . Our main theorem classifies the almost isotropic simply connected Kähler manifolds, proving that those of dimension {d=2n\geqslant 4} are either isometric to complex projective space or complex hyperbolic space or are totally geodesically foliated by leaves isometric to {{\mathbb{C}}^{n-1}} . 

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