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  • The bounded height conjecture for semiabelian varieties
    Compos. Math. (IF 1.2) Pub Date : 2020-06-30
    Lars Kühne

    The bounded height conjecture of Bombieri, Masser, and Zannier states that for any sufficiently generic algebraic subvariety of a semiabelian $\overline{\mathbb{Q}}$ -variety $G$ there is an upper bound on the Weil height of the points contained in its intersection with the union of all algebraic subgroups having (at most) complementary dimension in  $G$ . This conjecture has been shown by Habegger

    更新日期:2020-06-30
  • Computing a categorical Gromov–Witten invariant
    Compos. Math. (IF 1.2) Pub Date : 2020-06-18
    Andrei Căldăraru; Junwu Tu

    We compute the $g=1$ , $n=1$ B-model Gromov–Witten invariant of an elliptic curve $E$ directly from the derived category $\mathsf{D}_{\mathsf{coh}}^{b}(E)$ . More precisely, we carry out the computation of the categorical Gromov–Witten invariant defined by Costello using as target a cyclic $\mathscr{A}_{\infty }$ model of $\mathsf{D}_{\mathsf{coh}}^{b}(E)$ described by Polishchuk. This is the first

    更新日期:2020-06-18
  • Homological mirror symmetry for higher-dimensional pairs of pants
    Compos. Math. (IF 1.2) Pub Date : 2020-06-18
    Yankı Lekili; Alexander Polishchuk

    Using Auroux’s description of Fukaya categories of symmetric products of punctured surfaces, we compute the partially wrapped Fukaya category of the complement of $k+1$ generic hyperplanes in $\mathbb{CP}^{n}$ , for $k\geqslant n$ , with respect to certain stops in terms of the endomorphism algebra of a generating set of objects. The stops are chosen so that the resulting algebra is formal. In the

    更新日期:2020-06-18
  • An exceptional Siegel–Weil formula and poles of the Spin L-function of $\text{PGSp}_{6}$
    Compos. Math. (IF 1.2) Pub Date : 2020-05-29
    Wee Teck Gan; Gordan Savin

    We show a Siegel–Weil formula in the setting of exceptional theta correspondence. Using this, together with a new Rankin–Selberg integral for the Spin L-function of $\text{PGSp}_{6}$ discovered by Pollack, we prove that a cuspidal representation of $\text{PGSp}_{6}$ is a (weak) functorial lift from the exceptional group $G_{2}$ if its (partial) Spin L-function has a pole at $s=1$ .

    更新日期:2020-05-29
  • Corrigendum: Around $\boldsymbol{\ell }$ -independence
    Compos. Math. (IF 1.2) Pub Date : 2020-05-29
    Bruno Chiarellotto; Christopher Lazda

    We correct the proof of the main $\ell$ -independence result of the above-mentioned paper by showing that for any smooth and proper variety over an equicharacteristic local field, there exists a globally defined such variety with the same ( $p$ -adic and $\ell$ -adic) cohomology.

    更新日期:2020-05-29
  • Shimura varieties at level $\unicode[STIX]{x1D6E4}_{1}(p^{\infty })$ and Galois representations
    Compos. Math. (IF 1.2) Pub Date : 2020-05-26
    Ana Caraiani; Daniel R. Gulotta; Chi-Yun Hsu; Christian Johansson; Lucia Mocz; Emanuel Reinecke; Sheng-Chi Shih

    We show that the compactly supported cohomology of certain $\text{U}(n,n)$ - or $\text{Sp}(2n)$ -Shimura varieties with $\unicode[STIX]{x1D6E4}_{1}(p^{\infty })$ -level vanishes above the middle degree. The only assumption is that we work over a CM field $F$ in which the prime $p$ splits completely. We also give an application to Galois representations for torsion in the cohomology of the locally symmetric

    更新日期:2020-05-26
  • Cuspidal cohomology of stacks of shtukas
    Compos. Math. (IF 1.2) Pub Date : 2020-05-14
    Cong Xue

    Let $G$ be a connected split reductive group over a finite field $\mathbb{F}_{q}$ and $X$ a smooth projective geometrically connected curve over $\mathbb{F}_{q}$ . The $\ell$ -adic cohomology of stacks of $G$ -shtukas is a generalization of the space of automorphic forms with compact support over the function field of $X$ . In this paper, we construct a constant term morphism on the cohomology of stacks

    更新日期:2020-05-14
  • Monoidal categorification and quantum affine algebras
    Compos. Math. (IF 1.2) Pub Date : 2020-04-27
    Masaki Kashiwara; Myungho Kim; Se-jin Oh; Euiyong Park

    We introduce and investigate new invariants of pairs of modules $M$ and $N$ over quantum affine algebras $U_{q}^{\prime }(\mathfrak{g})$ by analyzing their associated $R$ -matrices. Using these new invariants, we provide a criterion for a monoidal category of finite-dimensional integrable $U_{q}^{\prime }(\mathfrak{g})$ -modules to become a monoidal categorification of a cluster algebra.

    更新日期:2020-04-27
  • Characteristic directions of two-dimensional biholomorphisms
    Compos. Math. (IF 1.2) Pub Date : 2020-03-31
    Lorena López-Hernanz; Rudy Rosas

    We prove that for each characteristic direction $[v]$ of a tangent to the identity diffeomorphism of order $k+1$ in $(\mathbb{C}^{2},0)$ there exist either an analytic curve of fixed points tangent to $[v]$ or $k$ parabolic manifolds where all the orbits are tangent to $[v]$ , and that at least one of these parabolic manifolds is or contains a parabolic curve.

    更新日期:2020-04-20
  • Classification of algebraic solutions of irregular Garnier systems
    Compos. Math. (IF 1.2) Pub Date : 2020-04-14
    Karamoko Diarra; Frank Loray

    We prove that algebraic solutions of Garnier systems in the irregular case are of two types. The classical ones come from isomonodromic deformations of linear equations with diagonal or dihedral differential Galois group; we give a complete list in the rank-2 case (two indeterminates). The pull-back ones come from deformations of coverings over a fixed degenerate hypergeometric equation; we provide

    更新日期:2020-04-20
  • Local Rankin–Selberg integrals for Speh representations
    Compos. Math. (IF 1.2) Pub Date : 2020-04-13
    Erez M. Lapid; Zhengyu Mao

    We construct analogues of Rankin–Selberg integrals for Speh representations of the general linear group over a $p$ -adic field. The integrals are in terms of the (extended) Shalika model and are expected to be the local counterparts of (suitably regularized) global integrals involving square-integrable automorphic forms and Eisenstein series on the general linear group over a global field. We relate

    更新日期:2020-04-20
  • Unistructurality of cluster algebras
    Compos. Math. (IF 1.2) Pub Date : 2020-04-17
    Peigen Cao; Fang Li

    We prove that any skew-symmetrizable cluster algebra is unistructural, which is a conjecture by Assem, Schiffler and Shramchenko. As a corollary, we obtain that a cluster automorphism of a cluster algebra ${\mathcal{A}}({\mathcal{S}})$ is just an automorphism of the ambient field ${\mathcal{F}}$ which restricts to a permutation of the cluster variables of ${\mathcal{A}}({\mathcal{S}})$ .

    更新日期:2020-04-20
  • Lower bounds for Maass forms on semisimple groups
    Compos. Math. (IF 1.2) Pub Date : 2020-04-17
    Farrell Brumley; Simon Marshall

    Let $G$ be an anisotropic semisimple group over a totally real number field $F$ . Suppose that $G$ is compact at all but one infinite place $v_{0}$ . In addition, suppose that $G_{v_{0}}$ is $\mathbb{R}$ -almost simple, not split, and has a Cartan involution defined over $F$ . If $Y$ is a congruence arithmetic manifold of non-positive curvature associated with $G$ , we prove that there exists a sequence

    更新日期:2020-04-20
  • Motohashi’s fourth moment identity for non-archimedean test functions and applications
    Compos. Math. (IF 1.2) Pub Date : 2020-04-17
    Valentin Blomer; Peter Humphries; Rizwanur Khan; Micah B. Milinovich

    Motohashi established an explicit identity between the fourth moment of the Riemann zeta function weighted by some test function and a spectral cubic moment of automorphic $L$ -functions. By an entirely different method, we prove a generalization of this formula to a fourth moment of Dirichlet $L$ -functions modulo $q$ weighted by a non-archimedean test function. This establishes a new reciprocity

    更新日期:2020-04-20
  • Projective objects and the modified trace in factorisable finite tensor categories
    Compos. Math. (IF 1.2) Pub Date : 2020-03-26
    Azat M. Gainutdinov; Ingo Runkel

    For ${\mathcal{C}}$ a factorisable and pivotal finite tensor category over an algebraically closed field of characteristic zero we show: (1) ${\mathcal{C}}$ always contains a simple projective object; (2) if ${\mathcal{C}}$ is in addition ribbon, the internal characters of projective modules span a submodule for the projective $\text{SL}(2,\mathbb{Z})$ -action; (3) the action of the Grothendieck ring

    更新日期:2020-03-27
  • Stability in the high-dimensional cohomology of congruence subgroups
    Compos. Math. (IF 1.2) Pub Date : 2020-03-24
    Jeremy Miller; Rohit Nagpal; Peter Patzt

    We prove a representation stability result for the codimension-one cohomology of the level-three congruence subgroup of $\mathbf{SL}_{n}(\mathbb{Z})$ . This is a special case of a question of Church, Farb, and Putman which we make more precise. Our methods involve proving finiteness properties of the Steinberg module for the group $\mathbf{SL}_{n}(K)$ for $K$ a field. This also lets us give a new proof

    更新日期:2020-03-24
  • Relative cubulations and groups with a 2-sphere boundary
    Compos. Math. (IF 1.2) Pub Date : 2020-03-24
    Eduard Einstein; Daniel Groves

    We introduce a new kind of action of a relatively hyperbolic group on a $\text{CAT}(0)$ cube complex, called a relatively geometric action. We provide an application to characterize finite-volume Kleinian groups in terms of actions on cube complexes, analogous to the results of Markovic and Haïssinsky in the closed case.

    更新日期:2020-03-24
  • Counting Higgs bundles and type $A$ quiver bundles
    Compos. Math. (IF 1.2) Pub Date : 2020-02-27
    Sergey Mozgovoy; Olivier Schiffmann

    We prove a closed formula counting semistable twisted (or meromorphic) Higgs bundles of fixed rank and degree over a smooth projective curve of genus $g$ defined over a finite field, when the twisting line bundle degree is at least $2g-2$ (this includes the case of usual Higgs bundles). This yields a closed expression for the Donaldson–Thomas invariants of the moduli spaces of twisted Higgs bundles

    更新日期:2020-02-27
  • Gonality of dynatomic curves and strong uniform boundedness of preperiodic points
    Compos. Math. (IF 1.2) Pub Date : 2020-02-17
    John R. Doyle; Bjorn Poonen

    Fix $d\geqslant 2$ and a field $k$ such that $\operatorname{char}k\nmid d$ . Assume that $k$ contains the $d$ th roots of $1$ . Then the irreducible components of the curves over $k$ parameterizing preperiodic points of polynomials of the form $z^{d}+c$ are geometrically irreducible and have gonality tending to $\infty$ . This implies the function field analogue of the strong uniform boundedness conjecture

    更新日期:2020-02-18
  • Deformation spaces and normal forms around transversals
    Compos. Math. (IF 1.2) Pub Date : 2020-02-17
    Francis Bischoff; Henrique Bursztyn; Hudson Lima; Eckhard Meinrenken

    Given a manifold $M$ with a submanifold $N$ , the deformation space ${\mathcal{D}}(M,N)$ is a manifold with a submersion to $\mathbb{R}$ whose zero fiber is the normal bundle $\unicode[STIX]{x1D708}(M,N)$ , and all other fibers are equal to $M$ . This article uses deformation spaces to study the local behavior of various geometric structures associated with singular foliations, with $N$ a submanifold

    更新日期:2020-02-18
  • The algebraic dimension of compact complex threefolds with vanishing second Betti number
    Compos. Math. (IF 1.2) Pub Date : 2020-02-13
    Frédéric Campana; Jean-Pierre Demailly; Thomas Peternell

    We study compact complex three-dimensional manifolds with vanishing second Betti number. In particular, we show that a compact complex manifold homeomorphic to the six-dimensional sphere does carry any non-constant meromorphic function.

    更新日期:2020-02-13
  • Ordinary primes in Hilbert modular varieties
    Compos. Math. (IF 1.2) Pub Date : 2020-02-06
    Junecue Suh

    A well-known conjecture, often attributed to Serre, asserts that any motive over any number field has infinitely many ordinary reductions (in the sense that the Newton polygon coincides with the Hodge polygon). In the case of Hilbert modular cuspforms $f$ of parallel weight $(2,\ldots ,2)$ , we show how to produce more ordinary primes by using the Sato–Tate equidistribution and combining it with the

    更新日期:2020-02-07
  • On the $K(\unicode[STIX]{x1D70B},1)$ -problem for restrictions of complex reflection arrangements
    Compos. Math. (IF 1.2) Pub Date : 2020-01-20
    Nils Amend; Pierre Deligne; Gerhard Röhrle

    Let $W\subset \operatorname{GL}(V)$ be a complex reflection group and $\mathscr{A}(W)$ the set of the mirrors of the complex reflections in  $W$ . It is known that the complement $X(\mathscr{A}(W))$ of the reflection arrangement $\mathscr{A}(W)$ is a $K(\unicode[STIX]{x1D70B},1)$ space. For $Y$ an intersection of hyperplanes in $\mathscr{A}(W)$ , let $X(\mathscr{A}(W)^{Y})$ be the complement in $Y$

    更新日期:2020-02-06
  • Symplectic quotients have symplectic singularities
    Compos. Math. (IF 1.2) Pub Date : 2020-01-31
    Hans-Christian Herbig; Gerald W. Schwarz; Christopher Seaton

    Let $K$ be a compact Lie group with complexification $G$ , and let $V$ be a unitary $K$ -module. We consider the real symplectic quotient $M_{0}$ at level zero of the homogeneous quadratic moment map as well as the complex symplectic quotient, defined here as the complexification of $M_{0}$ . We show that if $(V,G)$ is $3$ -large, a condition that holds generically, then the complex symplectic quotient

    更新日期:2020-01-31
  • L-spaces, taut foliations, and graph manifolds
    Compos. Math. (IF 1.2) Pub Date : 2020-01-23
    Jonathan Hanselman; Jacob Rasmussen; Sarah Dean Rasmussen; Liam Watson

    If $Y$ is a closed orientable graph manifold, we show that $Y$ admits a coorientable taut foliation if and only if $Y$ is not an L-space. Combined with previous work of Boyer and Clay, this implies that $Y$ is an L-space if and only if $\unicode[STIX]{x1D70B}_{1}(Y)$ is not left-orderable.

    更新日期:2020-01-23
  • Matching of orbital integrals (transfer) and Roche Hecke algebra isomorphisms
    Compos. Math. (IF 1.2) Pub Date : 2020-01-21
    Bertrand Lemaire; Manish Mishra

    Let $F$ be a non-Archimedean local field, $G$ a connected reductive group defined and split over $F$ , and $T$ a maximal $F$ -split torus in $G$ . Let $\unicode[STIX]{x1D712}_{0}$ be a depth-zero character of the maximal compact subgroup $T$ of $T(F)$ . This gives by inflation a character $\unicode[STIX]{x1D70C}$ of an Iwahori subgroup $\unicode[STIX]{x2110}\subset T$ of $G(F)$ . From Roche [Types

    更新日期:2020-01-22
  • The homological projective dual of $\operatorname{Sym}^{2}\mathbb{P}(V)$
    Compos. Math. (IF 1.2) Pub Date : 2020-01-17
    Jørgen Vold Rennemo

    We study the derived category of a complete intersection $X$ of bilinear divisors in the orbifold $\operatorname{Sym}^{2}\mathbb{P}(V)$ . Our results are in the spirit of Kuznetsov’s theory of homological projective duality, and we describe a homological projective duality relation between $\operatorname{Sym}^{2}\mathbb{P}(V)$ and a category of modules over a sheaf of Clifford algebras on $\mathbb

    更新日期:2020-01-17
  • A Witt Nadel vanishing theorem for threefolds
    Compos. Math. (IF 1.2) Pub Date : 2020-01-13
    Yusuke Nakamura; Hiromu Tanaka

    In this paper, we establish a vanishing theorem of Nadel type for the Witt multiplier ideals on threefolds over perfect fields of characteristic larger than five. As an application, if a projective normal threefold over $\mathbb{F}_{q}$ is not klt and its canonical divisor is anti-ample, then the number of the rational points on the klt-locus is divisible by $q$ .

    更新日期:2020-01-13
  • Quantum mirrors of log Calabi–Yau surfaces and higher-genus curve counting
    Compos. Math. (IF 1.2) Pub Date : 2020-01-07
    Pierrick Bousseau

    Gross, Hacking and Keel have constructed mirrors of log Calabi–Yau surfaces in terms of counts of rational curves. Using $q$ -deformed scattering diagrams defined in terms of higher-genus log Gromov–Witten invariants, we construct deformation quantizations of these mirrors and we produce canonical bases of the corresponding non-commutative algebras of functions.

    更新日期:2020-01-07
  • Loose Engel structures
    Compos. Math. (IF 1.2) Pub Date : 2020-01-06
    Roger Casals; Álvaro del Pino; Francisco Presas

    This paper contributes to the study of Engel structures and their classification. The main result introduces the notion of a loose family of Engel structures and shows that two such families are Engel homotopic if and only if they are formally homotopic. This implies a complete $h$ -principle when auxiliary data is fixed. As a corollary, we show that Lorentz and orientable Cartan prolongations are

    更新日期:2020-01-06
  • Pin(2)-equivariant Seiberg–Witten Floer homology of Seifert fibrations
    Compos. Math. (IF 1.2) Pub Date : 2019-12-09
    Matthew Stoffregen

    We compute the $\text{Pin}(2)$ -equivariant Seiberg–Witten Floer homology of Seifert rational homology three-spheres in terms of their Heegaard Floer homology. As a result of this computation, we prove Manolescu’s conjecture that $\unicode[STIX]{x1D6FD}=-\bar{\unicode[STIX]{x1D707}}$ for Seifert integral homology three-spheres. We show that the Manolescu invariants $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}

    更新日期:2020-01-04
  • Belyi’s theorem in characteristic two
    Compos. Math. (IF 1.2) Pub Date : 2019-12-18
    Yusuke Sugiyama; Seidai Yasuda

    We prove an analogue of Belyi’s theorem in characteristic two. Our proof consists of the following three steps. We first introduce a new notion called pseudo-tameness for morphisms between curves over an algebraically closed field of characteristic two. Secondly, we prove the existence of a ‘pseudo-tame’ rational function by showing the vanishing of an obstruction class. Finally, we construct a tamely

    更新日期:2020-01-04
  • Cohomology of generalized configuration spaces
    Compos. Math. (IF 1.2) Pub Date : 2019-12-20
    Dan Petersen

    Let $X$ be a topological space. We consider certain generalized configuration spaces of points on $X$ , obtained from the cartesian product $X^{n}$ by removing some intersections of diagonals. We give a systematic framework for studying the cohomology of such spaces using what we call ‘twisted commutative dg algebra models’ for the cochains on $X$ . Suppose that $X$ is a ‘nice’ topological space, $R$

    更新日期:2020-01-04
  • Hausdorff dimension of divergent trajectories on homogeneous spaces
    Compos. Math. (IF 1.2) Pub Date : 2019-12-19
    Lifan Guan; Ronggang Shi

    For a one-parameter subgroup action on a finite-volume homogeneous space, we consider the set of points admitting divergent-on-average trajectories. We show that the Hausdorff dimension of this set is strictly less than the manifold dimension of the homogeneous space. As a corollary we know that the Hausdorff dimension of the set of points admitting divergent trajectories is not full, which proves

    更新日期:2020-01-04
  • Vanishing and comparison theorems in rigid analytic geometry
    Compos. Math. (IF 1.2) Pub Date : 2019-12-26
    David Hansen

    We prove a rigid analytic analogue of the Artin–Grothendieck vanishing theorem. Precisely, we prove (under mild hypotheses) that the geometric étale cohomology of any Zariski-constructible sheaf on any affinoid rigid space $X$ vanishes in all degrees above the dimension of $X$ . Along the way, we show that branched covers of normal rigid spaces can often be extended across closed analytic subsets,

    更新日期:2020-01-04
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