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  • Projective objects and the modified trace in factorisable finite tensor categories
    Compos. Math. (IF 1.301) 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 1.301) 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 1.301) 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 1.301) 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 1.301) 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 1.301) 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 1.301) 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 1.301) 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 1.301) 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 1.301) 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 1.301) 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 1.301) 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 1.301) 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 1.301) 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 1.301) 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 1.301) 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 1.301) 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 1.301) 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 1.301) 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 1.301) 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 1.301) 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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