Abstract In this follow-up paper we show that smooth Hodge-proper stacks over script upper O Subscript upper K O K $\mathcal O_K$ are double struck upper Q Subscript p Q p $\mathbb Q_p$ -locally acyclic: namely the natural map between étale double struck upper Q Subscript p Q p $\mathbb Q_p$ -cohomology of the algebraic and Raynaud generic fibers is an equivalence. This establishes the double struck upper Q Subscript p Q p $\mathbb Q_p$ -case of general conjectures made in D. Kubrak and A. Prikhodko [ p-adic Hodge theory for Artin stacks , Mem. Amer. Math. Soc. 304 (2024), 1174]. As a corollary, we get that if a smooth Artin stack over K has a smooth Hodge-proper model over script upper O Subscript upper K O K $\mathcal O_K$ , its double struck upper Q Subscript p Q p $\mathbb Q_p$ -étale cohomology is a crystalline Galois representation. We then also establish a truncated version of the above results in more general setting of smooth d -de Rham-proper stacks over script upper O Subscript upper K O K $\mathcal O_K$ : here we only require first d -de Rham cohomology groups be finitely generated over script upper O Subscript upper K O K $\mathcal O_K$ . As an application, we deduce a certain purity-type statement for étale double struck upper Q Subscript p Q p $\mathbb Q_p$ -cohomology of Raynaud generic fiber, as well as crystallinity of a first several étale cohomology groups in the presence of a Cohen–Macauley model over script upper O Subscript upper K O K $\mathcal O_K$ in the schematic setting.
Abstract We prove that the set of limit points of the set of all spectral gaps of closed arithmetic hyperbolic surfaces equals left bracket 0 comma one fourth right bracket [ 0 , 1 4 ] $[0,\frac{1}{4}]$ .
Résumé Colmez, Dospinescu et Nizioł démontrent que les seules représentations p -adiques de normal upper G normal a normal l left parenthesis double struck upper Q overbar Subscript p Baseline divided by double struck upper Q Subscript p Baseline right parenthesis G a l ( Q ¯ p / Q p ) $\mathrm{Gal}(\bar{{\mathbb{Q}}}_p/{\mathbb{Q}}_p)$ qui apparaissent dans la cohomologie étale p -adique de la tour de revêtements du demi-plan de Drinfeld sont les représentations cuspidales (i.e. potentiellement semi-stables, dont la représentation de Weil-Deligne associée est irréductible) de dimension 2, à poids de Hodge-Tate 0 et 1 et que leur multiplicité est donnée par la correspondance de Langlands p -adique. Nous étendons ce résultat en poids quelconque, en considérant la cohomologie étale p -adique à coefficients dans les puissances symétriques du système local universel sur la tour de Drinfeld. Une différence notable est que toutes les représentations potentiellement semi-stables de dimension 2 non cristabélines, pas seulement les cuspidales, apparaissent avec les multiplicités attendues. Le point clé est que les systèmes locaux que l’on considère sont particulièrement simples : ce sont des ‘opers isotriviaux’ sur une courbe. Nous donnons une recette pour calculer la cohomologie proétale de tels systèmes locaux à partir de la cohomologie de Hyodo-Kato de la courbe et du complexe de de Rham du fibré plat filtré associé au système local.
Abstract This note corrects two errors in the paper Symplectomorphisms and spherical objects in the conifold smoothing [Compositio Math. 160 (2024), 2738–2773]. Neither affects the main results of the paper (in particular, neither affects any result stated in the Introduction of the paper). We apologise to our readers for these inaccuracies.
Abstract Let upper K divided by k K / k $K/k$ be a finite Galois extension of global function fields. Let E be a Drinfeld module over k . We state and prove an equivariant refinement of Taelman’s analogue of the analytic class number formula for left parenthesis upper E comma upper K divided by k right parenthesis ( E , K / k ) $(E,K/k)$ , and derive explicit consequences for the Galois structure of the Taelman class group of E over K .
We establish the following family version of Habegger's bounded height theorem on abelian varieties [Habegger, Intersecting subvarieties of abelian varieties with algebraic subgroups of complementary dimension, Invent. Math. 176 (2009a), 405-447]: a locally closed subvariety of an abelian scheme with Gao's tth degeneracy locus [Gao, Generic rank of Betti map and unlikely intersections, Compositio Math. 156 (2020a), 2469-2509] removed, intersected with all flat group subschemes of relative dimension at most t, gives a set of bounded total height. Our main tools include the Ax-Schanuel theorem, and intersection theory of adelic line bundles as developed by Yuan and Zhang [Adelic line bundles on quasi-projective varieties, Annals of Mathematics Studies, vol. 221 (Princeton University Press, Princeton, NJ, 2026)]. As two applications, we generalize Silverman's specialization theorem [Silverman, Heights and the specialization map for families of abelian varieties, J. Reine Angew. Math. 342 (1983), 197-211] to a higher-dimensional base, and establish a bounded height result towards Zhang's ICM conjecture [Zhang, Small points and Arakelov theory, in Proceedings of the international congress of mathematicians, Vol. II (Berlin, 1998), Extra Vol. II (1998), 217-225].
Consider a quadratic polynomial $Q(\xi_{1},\ldots,\xi_{n})$ of independent Rademacher random variables $\xi_{1},\ldots,\xi_{n}$ . To what extent can $Q(\xi_{1},\ldots,\xi_{n})$ concentrate on a single value? This quadratic version of the classical Littlewood-Offord problem was popularised by Costello, Tao and Vu in their study of symmetric random matrices. In this paper, we obtain an essentially optimal bound for this problem, as conjectured by Nguyen and Vu. Specifically, if $Q(\xi_{1},\ldots,\xi_{n})$ 'robustly depends on at least m of the $\xi_{i}$ ' in the sense that there is no way to pin down the value of $Q(\xi_{1},\ldots,\xi_{n})$ by fixing values for fewer than m of the variables $\xi_{i}$ , then we have $\mathrm{Pr}[Q(\xi_{1},\ldots,\xi_{n})=0]\le O(1/\sqrt{m})$ . This also implies a similar result in the case where $\xi_{1},\ldots,\xi_{n}$ have arbitrary distributions. Our proof combines a number of ideas that may be of independent interest, including an inductive decoupling scheme that reduces quadratic anticoncentration problems to high-dimensional linear anticoncentration problems. Also, one application of our main result is the resolution of a conjecture of Alon, Hefetz, Krivelevich and Tyomkyn related to graph inducibility.
We study the space of traces associated with arbitrary full free products of unital, separable $C*$ -algebras. We show that, unless certain basic obstructions (which we fully characterize) occur, the space of traces always results in the same object: the Poulsen simplex, that is, the unique infinite-dimensional metrizable Choquet simplex whose extreme points are dense. Moreover, we show that whenever such a trace space is the Poulsen simplex, the extreme points are dense in the Wasserstein topology. Concretely for the case of groups, we find that, unless the trivial character is isolated in the space of characters, the space of traces of any free product of non-trivial countable groups is the Poulsen simplex. Our main technical contribution is a new perturbation result for pairs of von Neumann subalgebras $(M_{1},M_{2})$ of a tracial von Neumann algebra M, providing necessary conditions under which $M_{1}$ and a small unitary perturbation of $M_{2}$ generate a II $_{1}$ factor.
We provide a complete classification of when the homeomorphism group of a stable surface, normal upper Sigma Sigma $\Sigma$ , has the automatic continuity property: Any homomorphism from normal upper H normal o normal m normal e normal o left parenthesis normal upper Sigma right parenthesis H o m e o ( Sigma ) $\mathrm{Homeo}({\Sigma})$ to a separable group is necessarily continuous. This result descends to a classification of when the mapping class group of normal upper Sigma Sigma $\Sigma$ has the automatic continuity property. Towards this classification, we provide a general framework for proving automatic continuity for groups of homeomorphisms. Applying this framework, we also show that the homeomorphism group of any stable, second-countable Stone space has the automatic continuity property. Under the presence of stability, this answers two questions of Mann.
A classical argument was introduced by Khintchine in 1926 in order to exhibit the existence of totally irrational singular linear forms in two variables. This argument was subsequently revisited and extended by many authors. For instance, in 1959 Jarn & iacute;k used it to show that for $n \geqslant 2$ and for any non-increasing positive f there are totally irrational matrices $A \in M_{m,n}(\mathbb{R})$ such that for all large enough t there are $\mathbf{p} \in \mathbb{Z}<^>m, \mathbf{q} \in \mathbb{Z}<^>n \smallsetminus \{0\}$ with $\|\mathbf{q}\| \leqslant t$ and $\|A \mathbf{q} - \mathbf{p}\| \leqslant f(t)$ . We denote the collection of such matrices by $\operatorname{UA}<^>*_{m,n}(f)$ . We adapt Khintchine's argument to show that the sets $\operatorname{UA}<^>*_{m,n}(f)$ , and their weighted analogues $\operatorname{UA}<^>*_{m,n}(f, {\boldsymbol{\omega}})$ , intersect many manifolds and fractals, and have strong intersection properties. For example, we show that: (i) when $n \geqslant 2$ , the set $\bigcap_{{\boldsymbol{\omega}}} \operatorname{UA}<^>*(f, {\boldsymbol{\omega}}) $ , where the intersection is over all weights ${\boldsymbol{\omega}}$ , is non-empty, and moreover intersects many manifolds and fractals; (ii) for $n \geqslant 2$ , there are vectors in $\mathbb{R}<^>n$ which are simultaneously k-singular for every k, in the sense of Yu; and (iii) when $n \geqslant 3$ , $\operatorname{UA}<^>*_{1,n}(f) + \operatorname{UA}<^>*_{1,n}(f) =\mathbb{R}<^>n$ . We also obtain new bounds on the rate of singularity which can be attained by column vectors in analytic submanifolds of dimension at least 2 in $\mathbb{R}<^>n$ .
A Laurent polynomial in two variables is tempered if its edge polynomials are cyclotomic. Variation of coefficients leads to a family of smooth complete genus g curves carrying a canonical algebraic K_2-class over a g-dimensional base S, hence to an extension of admissible variations of MHS (or normal function) on S. We prove that the ℝ-split locus of this extension is finite. Consequently, the torsion locus of the normal function and the A-polynomial locus for the family of curves are also finite.
We introduce and explore the Uniform Izumi-Rees Property in Noetherian rings with applications to multiplicity theory and containment relationships among symbolic powers of ideals. As an application, we prove that if R is a normal domain essentially of finite type over a field, there exists a constant C such that for all prime ideals German p subset of or equal to German q element of normal upper S normal p normal e normal c left parenthesis upper R right parenthesis p subset of q is an element of S p e c ( R ) $\mathfrak{p}\subseteq \mathfrak{q}\in\mathrm{Spec}(R)$ , if German p subset of or equal to German q Superscript left parenthesis t right parenthesis p subset of q ( t ) $\mathfrak{p}\subseteq \mathfrak{q}<^>{(t)}$ , then for all n element of double struck upper N n is an element of N $n\in\mathbb{N}$ , there is a containment of symbolic powers German p Superscript left parenthesis upper C n right parenthesis Baseline subset of or equal to German q Superscript left parenthesis t n right parenthesis p ( C n ) subset of q ( t n ) $\mathfrak{p}<^>{(Cn)}\subseteq \mathfrak{q}<^>{(tn)}$ .
This paper is concerned with local cohomology sheaves on generalized flag varieties supported in closed Schubert varieties, which carry natural structures as (mixed Hodge) A black, stylized uppercase letter 'D' in a decorative serif/calligraphic font on a white background. $\mathcal{D}$ -modules. We employ Kazhdan-Lusztig theory and Saito's theory of mixed Hodge modules to describe a general strategy to calculate the simple composition factors, Hodge filtration, and weight filtration on these modules. Our main tool is the Grothendieck-Cousin complex, introduced by Kempf, which allows us to relate the local cohomology modules in question to parabolic Verma modules over the corresponding Lie algebra. We show that this complex underlies a complex of mixed Hodge modules, and is thus endowed with Hodge and weight filtrations. We execute this strategy to calculate the composition factors and weight filtration for Schubert varieties in the Grassmannian; in particular, showing that the weight filtration is controlled by the admissible augmented Dyck patterns of Raicu and Weyman. As an application, upon restriction to the opposite big cell, we recover the composition factors and weight filtration on local cohomology with support in generic determinantal varieties.
We study the geometry of Calabi-Yau conifold transitions. This deformation process is known to possibly connect a Kähler threefold to a non-Kähler threefold. We use balanced and Hermitian-Yang-Mills metrics to geometrize the conifold transition and show that the whole operation is continuous in the Gromov-Hausdorff topology.
For log canonical (lc) algebraically integrable foliations on Kawamata log terminal (klt) varieties, we prove the base-point-freeness theorem, the contraction theorem, and the existence of flips. The first result resolves a conjecture of Cascini and Spicer, while the latter two results strengthen a result of Cascini and Spicer by removing their assumption on the termination of flips. Moreover, we prove the existence of the minimal model program for lc algebraically integrable foliations on klt varieties and the existence of good minimal models or Mori fiber spaces for lc algebraically integrable foliations polarized by ample divisors on klt varieties. As a consequence, we show that $\mathbb{Q}$ -factorial klt varieties with lc algebraically integrable Fano foliation structures are Mori dream spaces. We also show the existence of a Shokurov-type polytope for lc algebraically integrable foliations.
We give a categorical formulation of the $p$-adic local Langlands correspondence for $\mathrm{GL}_2(\mathbb{Q}_p)$,as an embedding of the derived category of locally admissible representations into the category of Ind-coherent sheaves on the moduli stack of two-dimensional representations of $\mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$. Moreover, we relate our version of the $p$-adic local Langlands correspondence for $\mathrm{GL}_2(\mathbb{Q}_p)$ to the cohomology of modular curves through a local-global compatibility formula.
Given a link of a normal surface singularity with its canonical contact structure, we compare the collection of its Stein fillings to its Milnor fillings (that is, Milnor fibers of possible smoothings). We prove that, unlike Stein fillings, Milnor fillings of a given link have bounded topology; for links of sandwiched singularities, we further establish that there are only finitely many Milnor fillings. We discuss some other obstructions for a Stein filling to be represented by a Milnor fiber, and for various types of singularities, including simple classes like cusps and triangle singularities, we produce Stein fillings that do not come from Milnor fibers or resolutions. Meanwhile, we discover that there are many contact 3-manifolds that admit infinitely many Stein fillings but do not admit arbitrarily large ones.
Let k be a perfect field of characteristic p and W(k) its ring of Witt vectors. We construct an equivalence of categories between the full subcategory of the derived category of quasi-coherent sheaves on the syntomification of W(k) spanned by objects whose Hodge-Tate weights are between [0,p-2] and an appropriate derived category of Fontaine-Laffaille modules.
Kreck proved that two 2q-manifolds are stably diffeomorphic if and only if they admit normally bordant normal (q-1)-smoothings over the same normal (q-1)-type (B, xi). We show that 'stably diffeomorphic' can be replaced by 'diffeomorphic' if the normal smoothings have isomorphic Q-forms (consisting of the intersection form of the manifold and the induced homomorphism on H-q), when the manifolds are simply-connected, q = 2k is even and H-q(B) is free. This proves a special case of Crowley's Q-form conjecture. The basis of the proof is the construction of an extended surgery obstruction associated to a normal bordism. As an application, we identify the inertia group of a (2k-1)-connected 4k-manifold with the kernel of a certain bordism map. By the calculations of Senger and Zhang and earlier results, these kernels are now known in all cases. For k = 2, 4, the combination of these results determines the inertia groups. We also obtain, for a simply-connected 4k-manifold M with normal (2k-1)-type (B, xi) such that H-2k(B) is free, an algebraic description of the stable class of M, that is, the set of diffeomorphism classes of manifolds stably diffeomorphic to M. Using this description, we explicitly compute the stable class of manifolds M with rank-2 hyperbolic intersection form.
