We give an effective realization of Kawamata’s famous bound for Fano $3$-folds. Our result is a list of $39,\!550$ candidate Hilbert series P for Fano $3$-folds satisfying the Bogomolov stability inequality. For each P, there may or may not be a Fano $3$-fold X that realizes P, but even so we give a set of weights that we expect to be among those of a minimal generating set of the anticanonical ring of any such X. This provides a crude first approximation to the ongoing classification of Fano $3$-folds. We explain in detail what this data means, the important nuances that current known examples reveal about it, and how it relates to a range of different classification programs.

We prove analogs of the Dirichlet theorem on arithmetic progressions and the Siegel–Walfisz theorem for the digital reverses of primes in arbitrary bases, which we obtained only for large bases in a previous paper. The proof is based on a generalization of the result of Martin–Mauduit–Rivat (2014) on exponential sums over primes with so-called “digital” functions.

We give a bound on the dimension of the linear automorphism group of a projective variety $Z \subset {\mathbb P} V$ in terms of its fundamental forms at a general point. Moreover, we show that the bound is achieved precisely when $Z \subset {\mathbb P} V$ is projectively equivalent to an Euler-symmetric variety. As a by-product, we determine the Lie algebra of infinitesimal automorphisms of an Euler-symmetric variety and also obtain a rigidity result on the specialization of an Euler-symmetric variety preserving the isomorphism type of the fundamental forms.

We show that the double dual EPW sextic associated with a strongly smooth Gushel–Mukai surface can be realized as a moduli space of semistable objects on its bounded derived category. Also, we observe that the double dual EPW surface associated with a special Gushel–Mukai threefold can be realized as a moduli space of semistable objects on its Kuznetsov component. Then we discuss extensions of our main results to double EPW sextics and double EPW surfaces and a refinement of a statement of Bayer and Perry about Gushel–Mukai threefolds with equivalent Kuznetsov components, under a mild assumption.

We prove the uniqueness of general Bessel models for $\mathrm {GSpin}$ groups over a local field of characteristic zero. The proof is to reduce it to the spherical case, which has been proved by Emory and Takeda in the non-archimedean case and by Emory, Kim, and Maiti in the archimedean case.

It is not known whether the realization part of the s-cobordism theorem holds for smooth $4$-manifolds, nor whether every pair of smoothly h-cobordant 4-manifolds is also smoothly s-cobordant. We provide some new conditions under which these questions admit a positive answer. We also give conditions under which the “standard” method to construct an h-cobordism with specified torsion cannot work.

We explore Liouville’s theorem and the strong Liouville property (SLP) for harmonic functions on Riemannian cones and surfaces. Our approach recasts the classical Liouville property in terms of the growth of radial eigenfunctions (in the case of manifolds with rotational symmetry), allowing us to recover and sharpen known results under minimal assumptions. We provide explicit estimates for the slowest-growing non-constant harmonic functions on cones and surfaces and construct examples where doubling fails but Liouville and SLP still hold. Finally, we prove a nonlinear Liouville theorem for p-subharmonic functions, $p\geq 2$, under curvature bounds, in complete Riemannian surfaces with a pole which simultaneously recover Milnor’s and Cheng–Yau’s theorems as particular cases. This result appears to be new and suggests a unified geometric perspective on linear and nonlinear Liouville phenomena.

It is shown that a quiver is left Noetherian if and only if the category of quiver representations in any locally Noetherian abelian category is again locally Noetherian. Here, locally Noetherian means that any object is the directed union of its Noetherian subobjects. For a quiver to be left Noetherian means that the left ideals of paths starting at any fixed vertex satisfy the ascending chain condition. The proof generalizes to representations of any small category that admits a Gröbner enrichment.

We investigate the theta correspondence of cusp forms for the dual pair $(O^*(4),\mathrm {Sp}(1,1))$ originally introduced by Tsuneo Arakawa in the non-adelic setting. We call this Arakawa lifting. In this article, reformulating the theta correspondence in the adelic setting, we provide commutation relations of Hecke operators satisfied by the Arakawa lifting at all non-archimedean places, which is referred to as Eichler commutation relations for classical modular forms. Their archimedean analog is also given in terms of reproducing kernels.

For $n\geq 4$, let X be a complex smooth Fano n-fold whose minimal anticanonical degree of non-free rational curves on X is at least $n-2$. We classify extremal contractions of such varieties. As an application, we obtain a classification of Fano fourfolds with pseudoindex and Picard number greater than one. Combining this result with previous results, we complete the classification of smooth Fano n-folds with pseudoindex at least $n-2$ and Picard number greater than one. This can be seen as a generalization of various previous results. We also discuss the relations between pseudoindex and other invariants of Fano varieties.

Let G be the real points of a connected semisimple simply connected algebraic group over ${\mathbb Q}$ such that G has a compact Cartan subgroup and let $\Gamma $ be a uniform lattice in G. Let $\widehat {G}_d$ denote the set of equivalence classes of unitary discrete series representations of G. We prove that for any finite subset of $\widehat {G}_d$ satisfying a certain condition, the associated finite set of discrete series multiplicities in $L^2(\Gamma \backslash G)$ determines all discrete series multiplicities in $L^2(\Gamma \backslash G)$. This allows us to obtain a refinement of the strong multiplicity one result for discrete series representations. As an application, we deduce that for two given levels, the equality of the dimensions of the spaces of cusp forms over a suitable finite set of weights implies the equality of the dimensions of the spaces of cusp forms for all weights.

In this article, we define “animated affinoid algebras” and prove some basic properties of them. Then we generalize the result of Mann (2022, A p-adic 6-functor formalism in rigid-analytic geometry) and Mikami (2024, Int. Math. Res. Not. IMRN, 7099–7128) (fppf-descent for discrete rings or affinoid algebras) to discrete animated rings or animated affinoid algebras.

Joyce structures are a class of geometric structures that first arose in relation to Donaldson–Thomas theory. There is a special class of examples, called class $S[A_1]$, whose underlying manifold parameterizes Riemann surfaces of some fixed genus equipped with a meromorphic quadratic differential with poles of fixed orders. We study two Joyce structures of this type using the isomonodromic systems associated with the Painlevé II and III$_3$ equations. We give explicit formulas for the Plebański functions of these Joyce structures and compute several associated objects, including their tau functions, which we explicitly relate to the corresponding Painlevé tau functions. We show that the behavior of the Joyce structure near the zero section can be studied analytically through poles of Painlevé equations. The systematic treatment gives a blueprint for the study of more general Joyce structures associated with meromorphic quadratic differentials on the Riemann sphere.

We show that if X is a regular complex projective threefold and Y is a very general surface of sufficiently high degree containing a thick infinitesimal neighborhood Z of a smooth curve C, then the restriction map $r: \textrm{Pic}\ X \to \textrm{Pic}\Y$ is an isomorphism. Our proof uses Moishezon’s Noether–Lefschetz theorem and some general tools of Hartshorne and Polini. We give the complete picture for all d when $X = {\mathbb P}^3$, C is a line, and $m=2$, using a classification of Urabe to handle the difficult case of quartic surfaces.

In this article, we extend our earlier unique continuation results of Pan and Zhang (2025, Comm. Anal. Geom. 33, 1975–2014) for the Schrödinger-type inequality $ |\bar \partial u| \le V|u|$ on a domain in $\mathbb C^n$, by removing the smoothness assumption on solutions $u = (u_1, \ldots , u_N)$. More specifically, we establish the unique continuation property for $W_{loc}^{1,1}$ solutions when the potential $V\in L_{loc}^p $, $ p>2n$, and for $W_{loc}^{1,2n+\epsilon }$ solutions when $V\in L_{loc}^{2n}$ with $N=1$ or $n = 2$. Although the unique continuation property fails in general if $V\in L_{loc}^{p}, p<2n$, we show that the property still holds for $W_{loc}^{1,1}$ solutions when $V $ is a small constant multiple of $ \frac {1}{|z|}$. The key idea that allows us to lower the regularity assumption is a careful analysis of how regularity and flatness are transferred to the sliced functions along complex radial directions.

We study properties of compactly supported p-adic pro-étale cohomology of smooth partially proper rigid analytic varieties. In particular, we prove a comparison theorem, in a stable range, with compactly supported syntomic cohomology, which is built from compactly supported Hyodo–Kato and $\mathbf {B}^+_{\mathrm {dR} }$-cohomologies. We derive from that a (limited version of a) fundamental diagram. This article should be thought of as a compactly supported counterpart of the series (Colmez and Nizioł, 2020, Münster J. Math. 13, 445–507; Colmez and Nizioł, 2025, J. Algebraic Geom. 34, 1–108; and Colmez and Nizioł, 2025, Duke Math. J. 174, 2203–2301) but, in particular, as a continuation of Colmez and Nizioł (2025).

Square-free powers of square-free monomial ideals relate homological properties of monomial ideals with combinatorial matching structures. Let $ I $ be any square-free monomial ideal, and let $ \mathcal {H}_I $ denote its associated hypergraph. We refine the concept of $ k $-admissible matching, originally introduced by Erey and Hibi for graphs, by defining a generalized version for arbitrary hypergraphs. This new notion allows us to establish a sharp lower bound for the Castelnuovo–Mumford regularity of the $ k\text {th} $ square-free power $ I^{[k]} $. When $ I $ is equigenerated, the bound is given in terms of a new combinatorial invariant, the $ k $-admissible matching number $ \mathrm {aim}(\mathcal {H}_I, k) $, provided $ I^{[k]} \neq 0 $. Surprisingly, for edge ideals $ I(G) $ of graphs, this gives the first general lower bound on the regularity of $ I(G)^{[k]}$. When $ G $ is a forest, our invariant coincides with the $ k $-admissible matching number of Erey and Hibi. Moreover, we prove that for any block graph $ G $, the bound becomes an exact formula, thereby generalizing the known regularity formula for forests. We conclude by proposing a conjecture on the regularity of square-free powers of edge ideals of chordal graphs.

We study the construction of a modular generalized Springer correspondence for a possibly disconnected complex reductive algebraic group.