Abstract Given a smooth proper family f : X → S $f:X\rightarrow S$ f colon upper X right arrow upper S , we study the (quasi)-periods of the fibres of f as (germs of) functions on S . We show that the field they generate has the same algebraic closure as that given by the flag variety coordinates parametrizing the corresponding Hodge filtration, together with their derivatives. Moreover, in the more general context of an arbitrary flat vector bundle, we determine the transcendence degree of the function field generated by the flat coordinates of algebraic sections. Our results are inspired by and generalize work of Bertrand–Zudilin.
Abstract In this paper we are going to compute the KW $ \mathrm {KW} $ upper K upper W -Euler classes for rank 2 vector bundles on the classifying stack B N $ \mathrm {B} \mathrm{N} $ normal upper B normal upper N , where N is the normaliser of the standard torus in SL 2 $\mathrm{SL}_2$ upper S upper L 2 and KW $\mathrm {KW}$ upper K upper W represents Balmer’s derived Witt groups. Using these computations we will recover, through a new and different strategy, the formulas previously obtained by Levine in Witt-sheaf cohomology. In order to obtain our results, we will prove Künneth formulas for products of GL n $\mathrm{GL}_n$ upper G upper L Subscript n ’s and SL n $\mathrm{SL}_n$ upper S upper L Subscript n ’s classifying spaces and we will develop from scratch the basic theory of twisted symplectic bundles with their associated twisted Borel classes in SL $\mathrm{SL}$ upper S upper L -oriented theories.
Given a smooth variety X over C $\mathbb {C}$ double struck upper C , a smooth divisor i : Y hooked right arrow X $i:Y\hookrightarrow X$ i colon upper Y right arrow with hook upper X and a global function f on X which vanishes on Y and on its critical locus, we compute the map induced on Hochschild homology by the pushforward functor i & lowast; : D b ( Y ) -> D a b s ( M F ( X , f ) ) $i_{\ast }:D<^>b(Y) o D<^>{abs}(MF(X,f))$ i Subscript asterisk Baseline colon upper D Superscript b Baseline left parenthesis upper Y right parenthesis right arrow upper D Superscript a b s Baseline left parenthesis upper M upper F left parenthesis upper X comma f right parenthesis right parenthesis in terms of the Hochschild-Kostant-Rosenberg isomorphisms.
Abstract Let E / F $E/F$ upper E divided by upper F be a quadratic extension of totally real number fields. We show that the generalized Hirzebruch–Zagier cycles arising from the associated Hilbert modular varieties can be put in p -adic families. As an application, using the theory of base change, we give a geometric construction of the multivariable p -adic adjoint L -function twisted by the Hecke character of E / F $E/F$ upper E divided by upper F , attached to Hida families of Hilbert modular forms over F .
Abstract We examine sets A $\mathscr A$ script upper A of natural numbers having the property that for some real number p ∈ ( 0 , 2 ) $p\in (0,2)$ p element of left parenthesis 0 comma 2 right parenthesis , one has the subconvex bound ∫ 0 1 | ∑ n ∈ A ∩ [ 1 , N ] e ( n α ) | p d α ≪ N − 1 | A ∩ [ 1 , N ] | p . $$\begin{align*}\int_0^1 \Big| \sum_{n\in \mathscr A\cap [1,N]}e(n\alpha)\Big|^p{\,\mathrm{d}} \alpha \ll N^{-1}|\mathscr A\cap [1,N]|^p. \end{align*}$$ We show that exponential sums over such sets satisfy inequalities analogous to Weyl’s inequality, and in many circumstances of the same strength as classical versions of Weyl’s bound. We also examine equidistribution of polynomials modulo 1 $1$ 1 in which the summands are restricted to these subconvex L p $L^p$ upper L Superscript p -sets. In addition, we describe applications to problems involving character sums and averages of arithmetic functions.
We consider a smooth fibration equipped with a flat complex vector bundle and a hypersurface cutting the fibration into two pieces. Our main result is a gluing formula relating the Bismut-Lott analytic torsion form of the whole fibration to that of each piece. This result solves a conjecture proposed at a conference in G & ouml;ttingen in 2003. This result also leads to a higher Cheeger-M & uuml;ller/Bismut-Zhang theorem. Our approach combines an adiabatic limit along the normal direction of the hypersurface and a Witten-type deformation on the flat vector bundle.
We continue our study of Ulam's measure problem. In contrast to our previous works, we shift our focus from measures stratified by their additivity, to measures stratified by their indecomposability. The breakthrough here is obtained by replacing the classical 'least' function associated with ideals by a two-dimensional 'last' function associated with walks on ordinals. Consequently, we obtain conditions under which a measure admits not just infinite pairwise disjoint families of positive sets, but in fact families of maximum possible size. As an application we solve a problem left open in Shelah's Cardinal Arithmetic book, proving that for every weakly inaccessible cardinal $\kappa $ , if there exists a stationary subset of $\kappa $ that does not reflect at regulars, then the strong Ramsey relation $\kappa rightarrow [\kappa ]<^>2_\kappa $ holds.
A closed Riemannian three-manifold (Y,g) equipped with a torsion spinc structure determines a family of Dirac operators {D-B} parametrized by a b(1)(Y )-dimensional torus T-Y. In this paper, we develop techniques to study how the topology of the locus K subset of T-Y corresponding to operators with non-trivial kernel (the three-dimensional analogue of the theta divisor of a Riemann surface) depends on the geometry of the metric. As a concrete example of our methods, we show that for any metric on the three-torus Y = T-3 for which the spectral gap lambda(1)* on co exact 1-forms is large, after a small perturbation of the family, the locus K is a two-sphere. While the result only involves linear operators, its proof relies on the non-linear analysis of the Seib erg-Witten equations. It follows from a more general understanding of transversality in the context of the monopole Floer homology of a torsion spin (c) three-manifold (Y,s) with a large lambda(1)*. When b(1) > 0, this gives rise to a very rich setup and we discuss a framework to describe explicitly in certain situations the Floer homology groups of (Y,s) in terms of the topology of the family of Dirac operators {D-B}.
We consider induced representations $\operatorname {\mathrm {Ind}}_{\mathrm {P}(F)}<^>{\operatorname {\mathrm {G}}(F)} \pi $ , where $\mathrm {P}$ is a maximal parabolic subgroup of a reductive group $\operatorname {\mathrm {G}}$ over a p-adic field F, and $(\pi , V)$ is a unitary supercuspidal representation of $\operatorname {\mathrm {M}}(F)$ , $\operatorname {\mathrm {M}}$ being some Levi subgroup of $\mathrm {P}$ . Imposing a certain 'Heisenberg parabolic subgroup' assumption on $\mathrm {P}$ , we apply the method of Goldberg, Shahidi and Spallone to obtain an expression for a certain constant $R(\tilde {\pi })$ , which captures the residue of a family $s \mapsto A(s, \pi , w_0)$ of intertwining operators associated to this situation, in terms of harmonic analysis on the twisted Levi subgroup $\tilde {\operatorname {\mathrm {M}}}(F) := \operatorname {\mathrm {M}}(F) w_0$ . For $\operatorname {\mathrm {G}}$ absolutely almost simple and simply connected of type $G_2$ or $D_4$ (resp., $B_3$ ), and $\mathrm {P}$ satisfying the 'Heisenberg' condition, if the central character of $\pi $ is nontrivial (resp., trivial) on $\operatorname {\mathrm {A}}_{\operatorname {\mathrm {M}}}(F)$ , where $\operatorname {\mathrm {A}}_{\operatorname {\mathrm {M}}}$ is the connected centre of $\operatorname {\mathrm {M}}$ , our formula for $R(\tilde {\pi })$ can be rewritten in terms of the Langlands parameter of $\pi $ , in the spirit of a prediction of Arthur. For the same collection of $\operatorname {\mathrm {G}}$ and $\mathrm {P}$ , when these central character conditions are not satisfied, Arthur's prediction combined with our formula for $R(\tilde {\pi })$ suggests a harmonic analytic formula for a product of one or two $\gamma $ -factors associated to the situation.
For $r\geq 3$ and $g= \frac {r(r+1)}{2}$ , we study the Prym-Brill-Noether variety $V<^>r(C,\eta )$ associated to Prym curves $[C,\eta ]$ . The locus $\mathcal {R}_g<^>r$ in $\mathcal {R}_g$ parametrizing Prym curves $(C, \eta )$ with nonempty $V<^>r(C,\eta )$ is a divisor. We compute some key coefficients of the class $[\overline {\mathcal {R}}_g<^>r]$ in $\mathrm {Pic}_{\mathbb {Q}}(\overline {\mathcal {R}}_g)$ . Furthermore, we examine a strongly Brill-Noether divisor in $\overline {\mathcal {M}}_{g-1,2}$ : we show its irreducibility and compute some of its coefficients in $\mathrm {Pic}_{\mathbb {Q}}(\overline {\mathcal {M}}_{g-1,2})$ . As a consequence of our results, the moduli space $\mathcal {R}_{14,2}$ is of general type.
In this paper, we study the cohomology of the unitary unramified PEL Rapoport-Zink space of signature $(1,n-1)$ at hyperspecial level. Our method revolves around the spectral sequence associated to the open cover by the analytical tubes of the closed Bruhat-Tits strata in the special fiber, which were constructed by Vollaard and Wedhorn. The cohomology of these strata, which are isomorphic to generalized Deligne-Lusztig varieties, has been computed in an earlier work. This spectral sequence allows us to prove the semisimplicity of the Frobenius action and the non-admissibility of the cohomology in general. Via p-adic uniformization, we relate the cohomology of the Rapoport-Zink space to the cohomology of the supersingular locus of a Shimura variety with no level at p. In the case $n=3$ or $4$ , we give a complete description of the cohomology of the supersingular locus in terms of automorphic representations.
Consider a Berkovich space over a good Banach ring and the relative projective line over it. (It is a space whose fibers are projective lines over different complete valued fields.) For each polarized endomorphism of this line, we prove that the family of equilibrium measures associated to the restrictions of the endomorphism to the fibers is continuous. The result holds, in particular, when the Banach ring is a complete valued field, a hybrid field, a complete discrete valuation ring, or the ring of integers of a number field.
Let M be a pinched negatively curved Riemannian orbifold, whose fundamental group has torsion of order 2. Generalizing results of Sarnak and Erlandsson-Souto for constant curvature oriented surfaces, and with very different techniques, we give an asymptotic counting result on the number of strongly reversible periodic orbits of the geodesic flow in M, and prove their equidistribution towards the Bowen-Margulis measure. The result is proved in the more general setting with weights coming from thermodynamic formalism, and also in the analogous setting of graphs of groups with 2-torsion. We give new examples in real hyperbolic Coxeter groups, complex hyperbolic orbifolds and graphs of groups.
For a finite extension $F$ of $\mathbb{Q}_p$ and $n \geq 1$, we show that the category of Lubin-Tate bundles on the $(n-1)$-dimensional Drinfeld symmetric space is equivalent to the category of finite-dimensional smooth representations of the group of units of the division algebra of invariant $1/n$ over $F$.
For a complete discrete valuation field $K$, we show that one may always glue a separated formal algebraic space $\mathfrak{X}$ over $\mathcal{O}_K$ to a separated algebraic space $U$ over $K$ along an open immersion of rigid spaces $j\colon \mathfrak{X}^{\rm rig}\to U^{\rm an}$, producing a separated algebraic space $X$ over $\mathcal{O}_K$. This process gives rise to an equivalence between such `gluing triples' $(U,\mathfrak{X},j)$ and separated algebraic spaces $X$ over $\mathcal{O}_K$, which one might interpret as a version of the Beauville--Laszlo theorem for algebraic spaces rather than coherent sheaves. Moreover, an analogous equivalence exists over any excellent base. Examples due to Matsumoto imply that the result of such a gluing might be a genuine algebraic space (not a scheme) even if $U$ and the special fiber of $\mathfrak{X}$ are projective. The proof is a combination of Nagata compactification theorem for algebraic spaces and of Artin's contraction theorem. We give multiple examples and applications of this idea.
In recent years, b-symplectic manifolds have emerged as important objects in symplectic geometry. These manifolds are Poisson manifolds that exhibit symplectic behaviour away from a distinguished hypersurface, where the symplectic form degenerates in a controlled manner. Inspired by this rich landscape, E-structures were introduced by Nest and Tsygan in [NT01] as a comprehensive framework for exploring generalizations of b-structures. This paper initiates a deeper investigation into their Poisson facets, building on foundational work by [MS21]. We also examine the closely related concept of almost regular Poisson manifolds, as studied in [AZ17], which reveals a natural Poisson groupoid associated with these structures.In this article, we investigate the intricate relationship between E-structures and almost regular Poisson structures. Our comparative analysis not only scrutinizes their Poisson properties but also offers explicit formulae for the Poisson structure on the Poisson groupoid associated to the E-structures as both Poisson manifolds and singular foliations. In doing so, we reveal an interesting link between the existence of commutative frames and Darboux-Carath & eacute;odory-type expressions for the relevant structures.
We introduce a notion of stratification for rigidly-compactly generated tensor-triangulated categories relative to the homological spectrum and develop the fundamental features of this theory. In particular, we demonstrate that it exhibits excellent descent properties. In conjunction with Balmer's Nerves of Steel conjecture, we conclude that classical stratification also admits a general form of descent. This gives a uniform treatment of several recent stratification results and provides a complete answer to the question: When does stratification descend? As a new application, we extend earlier work on the tensor triangular geometry of equivariant module spectra from finite groups to compact Lie groups.
We study the rationality of some geometrically rational three-dimensional conic and quadric surface bundles, defined over the reals and more general real closed fields, for which the real locus is connected and the intermediate Jacobian obstructions to rationality vanish. We obtain both negative and positive results, using unramified cohomology and birational rigidity techniques, as well as concrete rationality constructions.
The integral identity conjecture of Kontsevich and Soibelman plays an important role in proving the existence of motivic Donaldson-Thomas invariants for three-dimensional noncommutative Calabi-Yau manifolds. There are a number of different formulations of this conjecture in different contexts, and accordingly, there are corresponding solutions to them. The methods devoted to solving this conjecture are diverse, ranging from script l $\ell $ & ell; -adic cohomology of rigid analytic varieties to Hrushovski-Kazhdan motivic integration and motivic Fubini theorem for tropicalization maps,... In [Ivo24], Ivorra deduces a functorial version of the integral identity in the motivic stable homotopy categories of schemes, from the Braden hyperbolic localization theorem. This functorial version concerns Ayoub's nearby cycles functor associated with a upper G Subscript m $\mathbb {G}_m$ G m -equivariant function f colon double struck upper V left parenthesis script upper E right parenthesis long right arrow upper A Superscript 1 $f \colon \mathbb {V}(\mathcal {E}) \longrightarrow \mathbb {A}<^>1$ f : V ( E ) -> A 1 on a vector bundle double struck upper V left parenthesis script upper E right parenthesis $\mathbb {V}(\mathcal {E})$ V ( E ) over a field of characteristic zero. In the present work, we follow the functorial approach from [Ivo24] and extend the scope of the original conjecture by Kontsevich and Soibelman by studying more generally the case of upper G Subscript m $\mathbb {G}_m$ G m -equivariant functions on algebraic S-spaces with a tau $ au $ tau -locally linearizable action of upper G Subscript m $\mathbb {G}_m$ G m over a noetherian base scheme S.
In the Morel-Voevodsky motivic stable homotopy category of a quasi-compact quasi-separated scheme S, several candidates exist for a motivic spectrum representing hermitian K-theory. This note shows that the cellular absolute motivic spectrum constructed in the thesis of the first author via the geometry of orthogonal and hyperbolic Grassmannians over any scheme coincides with the motivic ring spectrum constructed recently by Calmès, Harpaz, and Nardin.
