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1. Stabilization of the cohomology of thickenings

   Bhatt, Bhargav; Blickle, Manuel; Lyubeznik, Gennady; Singh, Anurag; Zhang, Wenliang. (April 2019, American journal of mathematics)

   For a local complete intersection subvariety $X = V (I)$ in $P^n$ over a field of characteristic zero, we show that, in cohomological degrees smaller than the codimension of the singular locus of $$X$$, the cohomology of vector bundles on the formal completion of $P^n$ along $$X$$ can be effectively computed as the cohomology on any sufficiently high thickening $$X_t = V (I^t)$$; the main ingredient here is a positivity result for the normal bundle of $$X$$. Furthermore, we show that the Kodaira vanishing theorem holds for all thickenings $$X_t$$ in the same range of cohomological degrees; this extends the known version of Kodaira vanishing on $$X$$, and the main new ingredient is a version of the Kodaira- Akizuki-Nakano vanishing theorem for $$X$$, formulated in terms of the cotangent complex. 

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2. Eulerianity of Fourier coefficients of automorphic forms

   https://doi.org/10.1090/ert/565  

   Gourevitch, Dmitry; Gustafsson, Henrik; Kleinschmidt, Axel; Persson, Daniel; Sahi, Siddhartha (June 2021, Representation Theory of the American Mathematical Society)

   We study the question of Eulerianity (factorizability) for Fourier coefficients of automorphic forms, and we prove a general transfer theorem that allows one to deduce the Eulerianity of certain coefficients from that of another coefficient. We also establish a ‘hidden’ invariance property of Fourier coefficients. We apply these results to minimal and next-to-minimal automorphic representations, and deduce Eulerianity for a large class of Fourier and Fourier–Jacobi coefficients. In particular, we prove Eulerianity for parabolic Fourier coefficients with characters of maximal rank for a class of Eisenstein series in minimal and next-to-minimal representations of groups of ADE-type that are of interest in string theory. 

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3. Torsion Invariants of complexes of groups

   https://doi.org/10.1215/00127094-2023-0024  

   Okun, Boris; Schreve, K (February 2024, Duke)

   Hain, Richard (Ed.)

   Suppose a residually finite group G acts cocompactly on a contractible complex with strict fundamental domain Q, where the stabilizers are either trivial or have normal Z-subgroups. Let dQ be the subcomplex of Q with nontrivial stabilizers. Our main result is a computation of the homology torsion growth of a chain of finite index normal subgroups of G. We show that independent of the chain, the normalized torsion limits to the torsion of dQ shifted a degree. Under milder assumptions of acyclicity of nontrivial stabilizers, we show similar formulas for the mod p-homology growth. We also obtain formulas for the universal and the usual L^2-torsion of G in terms of the torsion of stabilizers and topology of dQ. In particular, we get complete answers for right-angled Artin groups, which shows that they satisfy a torsion analogue of Lück’s approximation theorem. 

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4. Equivalence of codes for countable sets of reals

   https://doi.org/10.4153/S0008439520000661  

   Chan, William (September 2021, Canadian Mathematical Bulletin)

   null (Ed.)

   Abstract A set $$U \subseteq {\mathbb {R}} \times {\mathbb {R}}$$ is universal for countable subsets of $${\mathbb {R}}$$ if and only if for all $$x \in {\mathbb {R}}$$ , the section $$U_x = \{y \in {\mathbb {R}} : U(x,y)\}$$ is countable and for all countable sets $$A \subseteq {\mathbb {R}}$$ , there is an $$x \in {\mathbb {R}}$$ so that $$U_x = A$$ . Define the equivalence relation $$E_U$$ on $${\mathbb {R}}$$ by $$x_0 \ E_U \ x_1$$ if and only if $$U_{x_0} = U_{x_1}$$ , which is the equivalence of codes for countable sets of reals according to U . The Friedman–Stanley jump, $=^+$ , of the equality relation takes the form $$E_{U^*}$$ where $U^*$ is the most natural Borel set that is universal for countable sets. The main result is that $=^+$ and $$E_U$$ for any U that is Borel and universal for countable sets are equivalent up to Borel bireducibility. For all U that are Borel and universal for countable sets, $$E_U$$ is Borel bireducible to $=^+$ . If one assumes a particular instance of $$\mathbf {\Sigma }_3^1$$ -generic absoluteness, then for all $$U \subseteq {\mathbb {R}} \times {\mathbb {R}}$$ that are $$\mathbf {\Sigma }_1^1$$ (continuous images of Borel sets) and universal for countable sets, there is a Borel reduction of $=^+$ into $$E_U$$ . 

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5. Free resolutions of orbit closures of Dynkin quivers

   https://doi.org/10.1090/tran/7697  

   Lorincz, Andras; Weyman, Jerzy (May 2019, Transactions of the American Mathematical Society)

   We use the Kempf-Lascoux-Weyman geometric technique in order to determine the minimal free resolutions of some orbit closures of quivers. As a consequence, we obtain that for Dynkin quivers orbit closures of 1-step representations are normal with rational singularities. For Dynkin quivers of type $$ \mathbb{A}$$, we describe explicit minimal generators of the defining ideals of orbit closures of 1-step representations. Using this, we provide an algorithm for type $$ \mathbb{A}$$ quivers for describing an efficient set of generators of the defining ideal of the orbit closure of any representation. 

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