The construction problem for Hodge numbers modulo an integer in positive characteristic

Serre Duality ◽

Hodge Numbers ◽

Abstract Let k be an algebraically closed field of positive characteristic. For any integer $m\ge 2$ , we show that the Hodge numbers of a smooth projective k-variety can take on any combination of values modulo m, subject only to Serre duality. In particular, there are no non-trivial polynomial relations between the Hodge numbers.

THE LOEWY STRUCTURE OF -VERMA MODULES OF SINGULAR HIGHEST WEIGHTS

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748015000274 ◽

2015 ◽

Vol 16 (4) ◽

pp. 887-898

Author(s):

Noriyuki Abe ◽

Masaharu Kaneda

Keyword(s):

Algebraic Group ◽

Maximal Torus ◽

Positive Characteristic ◽

Closed Field ◽

Verma Modules ◽

Reductive Algebraic Group ◽

Loewy Length ◽

Frobenius Kernel ◽

Let $G$ be a reductive algebraic group over an algebraically closed field of positive characteristic, $G_{1}$ the Frobenius kernel of $G$, and $T$ a maximal torus of $G$. We show that the parabolically induced $G_{1}T$-Verma modules of singular highest weights are all rigid, determine their Loewy length, and describe their Loewy structure using the periodic Kazhdan–Lusztig $P$- and $Q$-polynomials. We assume that the characteristic of the field is sufficiently large that, in particular, Lusztig’s conjecture for the irreducible $G_{1}T$-characters holds.

New directions in the Minimal Model Program

Bollettino dell Unione Matematica Italiana ◽

10.1007/s40574-020-00250-9 ◽

2020 ◽

Author(s):

Paolo Cascini

Keyword(s):

Minimal Model ◽

Positive Characteristic ◽

Closed Field ◽

Model Program ◽

Minimal Model Program ◽

New Directions ◽

Abstract We survey some recents developments in the Minimal Model Program. After an elementary introduction to the program, we focus on its generalisations to the category of foliated varieties and the category of varieties defined over any algebraically closed field of positive characteristic.

UN SCINDAGE DU MORPHISME DE FROBENIUS SUR L’ALGÈBRE DES DISTRIBUTIONS D’UN GROUPE RÉDUCTIF

The Quarterly Journal of Mathematics ◽

10.1093/qmathj/haz039 ◽

2019 ◽

Vol 71 (1) ◽

pp. 197-206

Author(s):

Michel Gros ◽

Kaneda Masaharu

Keyword(s):

Nous Avons ◽

Algebraic Group ◽

Positive Characteristic ◽

Closed Field ◽

Reductive Groups ◽

Frobenius Endomorphism ◽

Simply Connected ◽

Simplement Connexe ◽

Abstract Pour un groupe algébrique semi-simple simplement connexe sur un corps algébriquement clos de caractéristique positive, nous avons précédemment construit un scindage de l’endomorphisme de Frobenius sur son algèbre des distributions. Nous généralisons la construction au cas de des groupes réductifs connexes et en dégageons les corollaires correspondants. For a simply connected semisimple algebraic group over an algebraically closed field of positive characteristic we have already constructed a splitting of the Frobenius endomorphism on its algebra of distributions. We generalize the construction to the case of general connected reductive groups and derive the corresponding corollaries.

On a purely inseparable analogue of the Abhyankar conjecture for affine curves

Compositio Mathematica ◽

10.1112/s0010437x18007194 ◽

2018 ◽

Vol 154 (8) ◽

pp. 1633-1658

Author(s):

Shusuke Otabe

Keyword(s):

Fundamental Group ◽

Positive Characteristic ◽

Smooth Curve ◽

Group Scheme ◽

Partial Answer ◽

Closed Field ◽

Affine Curves ◽

Finite Quotients ◽

Let$U$be an affine smooth curve defined over an algebraically closed field of positive characteristic. The Abhyankar conjecture (proved by Raynaud and Harbater in 1994) describes the set of finite quotients of Grothendieck’s étale fundamental group$\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(U)$. In this paper, we consider a purely inseparable analogue of this problem, formulated in terms of Nori’s profinite fundamental group scheme$\unicode[STIX]{x1D70B}^{N}(U)$, and give a partial answer to it.

Oort groups and lifting problems

Compositio Mathematica ◽

10.1112/s0010437x08003515 ◽

2008 ◽

Vol 144 (4) ◽

pp. 849-866 ◽

Cited By ~ 12

Author(s):

T. Chinburg ◽

R. Guralnick ◽

D. Harbater

Keyword(s):

Finite Groups ◽

Characteristic Zero ◽

Positive Characteristic ◽

Closed Field ◽

Projective Curve ◽

Smooth Projective Curve ◽

Cyclic Groups ◽

Characteristic Two ◽

AbstractLet k be an algebraically closed field of positive characteristic p. We consider which finite groups G have the property that every faithful action of G on a connected smooth projective curve over k lifts to characteristic zero. Oort conjectured that cyclic groups have this property. We show that if a cyclic-by-p group G has this property, then G must be either cyclic or dihedral, with the exception of A4 in characteristic two. This proves one direction of a strong form of the Oort conjecture.

The construction problem for Hodge numbers modulo an integer in positive characteristic—ERRATUM

Forum of Mathematics Sigma ◽

10.1017/fms.2020.67 ◽

2021 ◽

Vol 9 ◽

Author(s):

Remy van Dobben de Bruyn ◽

Matthias Paulsen

Keyword(s):

Positive Characteristic ◽

Construction Problem ◽

Hodge Numbers

Classification of one-dimensional superattracting germs in positive characteristic

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2014.35 ◽

2014 ◽

Vol 35 (7) ◽

pp. 2242-2268 ◽

Cited By ~ 2

Author(s):

MATTEO RUGGIERO

Keyword(s):

Positive Characteristic ◽

Closed Field ◽

One Dimensional ◽

Higher Dimensional ◽

Dimensional Version

We give a classification of superattracting germs in dimension $1$ over a complete normed algebraically closed field $\mathbb{K}$ of positive characteristic up to conjugacy. In particular, we show that formal and analytic classifications coincide for these germs. We also give a higher-dimensional version of some of these results.

The Hilbert series of the irreducible quotient of the polynomial representation of the rational Cherednik algebra of type An−1 in characteristic p for p|n − 1

Journal of Algebra and Its Applications ◽

10.1142/s0219498822501912 ◽

2021 ◽

pp. 2250191

Author(s):

Merrick Cai ◽

Daniil Kalinov

Keyword(s):

Hilbert Series ◽

Positive Characteristic ◽

Closed Field ◽

Simple Criterion ◽

Characteristic P ◽

Polynomial Representation ◽

Cherednik Algebra ◽

Rational Cherednik Algebra ◽

Polynomial Formula ◽

In this paper, we study the irreducible quotient [Formula: see text] of the polynomial representation of the rational Cherednik algebra [Formula: see text] of type [Formula: see text] over an algebraically closed field of positive characteristic [Formula: see text] where [Formula: see text]. In the [Formula: see text] case, for all [Formula: see text] we give a complete description of the polynomials in the maximal proper graded submodule [Formula: see text], the kernel of the contravariant form [Formula: see text], and subsequently find the Hilbert series of the irreducible quotient [Formula: see text]. In the [Formula: see text] case, we give a complete description of the polynomials in [Formula: see text] when the characteristic [Formula: see text] and [Formula: see text] is transcendental over [Formula: see text], and compute the Hilbert series of the irreducible quotient [Formula: see text]. In doing so, we prove a conjecture due to Etingof and Rains completely for [Formula: see text], and also for any [Formula: see text] and [Formula: see text]. Furthermore, for [Formula: see text], we prove a simple criterion to determine whether a given polynomial [Formula: see text] lies in [Formula: see text] for all [Formula: see text] with [Formula: see text] and [Formula: see text] fixed.

Fibrations with moving cuspidal singularities

Nagoya Mathematical Journal ◽

10.1017/s0027763000003597 ◽

1991 ◽

Vol 122 ◽

pp. 161-179 ◽

Cited By ~ 4

Author(s):

Yoshifumi Takeda

Keyword(s):

Algebraic Geometry ◽

Characteristic Zero ◽

Positive Characteristic ◽

Closed Field ◽

Projective Surface ◽

Projective Curve ◽

Smooth Projective Curve ◽

Smooth Projective Surface ◽

Almost All

Let f: V → C be a fibration from a smooth projective surface onto a smooth projective curve over an algebraically closed field k. In the case of characteristic zero, almost all fibres of f are nonsingular. In the case of positive characteristic, it is, however, known that there exist fibrations whose general fibres have singularities. Moreover, it seems that such fibrations often have pathological phenomena of algebraic geometry in positive characteristic (see M. Raynaud [7], W. Lang [4]).

Centers and Azumaya loci for finite W-algebras in positive characteristic

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004121000414 ◽

2021 ◽

pp. 1-32

Author(s):

BIN SHU ◽

YANG ZENG

Keyword(s):

Lie Algebra ◽

Nilpotent Element ◽

Positive Characteristic ◽

Irreducible Representations ◽

Maximal Dimension ◽

Closed Field ◽

Simple Lie Algebra ◽

Maximal Spectrum ◽

Smooth Locus

Abstract In this paper, we study the center Z of the finite W-algebra $${\mathcal{T}}({\mathfrak{g}},e)$$ associated with a semi-simple Lie algebra $$\mathfrak{g}$$ over an algebraically closed field $$\mathbb{k}$$ of characteristic p≫0, and an arbitrarily given nilpotent element $$e \in{\mathfrak{g}} $$ We obtain an analogue of Veldkamp’s theorem on the center. For the maximal spectrum Specm(Z), we show that its Azumaya locus coincides with its smooth locus of smooth points. The former locus reflects irreducible representations of maximal dimension for $${\mathcal{T}}({\mathfrak{g}},e)$$ .