scholarly journals The Griffiths bundle is generated by groups

2019 ◽  
Vol 375 (3-4) ◽  
pp. 1283-1305
Author(s):  
Wushi Goldring
Keyword(s):  
Line Bundle ◽  
Reductive Group ◽  
General Setting ◽  
Arbitrary Field ◽  
Period Map ◽  
General Fiber ◽  
Tate Group

Abstract First the Griffiths line bundle of a $$\mathbf {Q}$$ Q -VHS $${\mathscr {V}}$$ V is generalized to a Griffiths character $${{\,\mathrm{grif}\,}}(\mathbf {G}, \mu ,r)$$ grif ( G , μ , r ) associated to any triple $$(\mathbf {G}, \mu , r)$$ ( G , μ , r ) , where $$\mathbf {G}$$ G is a connected reductive group over an arbitrary field F, $$\mu \in X_*(\mathbf {G})$$ μ ∈ X ∗ ( G ) is a cocharacter (over $$\overline{F}$$ F ¯ ) and $$r:\mathbf {G}\rightarrow GL(V)$$ r : G → G L ( V ) is an F-representation; the classical bundle studied by Griffiths is recovered by taking $$F=\mathbf {Q}$$ F = Q , $$\mathbf {G}$$ G the Mumford–Tate group of $${\mathscr {V}}$$ V , $$r:\mathbf {G}\rightarrow GL(V)$$ r : G → G L ( V ) the tautological representation afforded by a very general fiber and pulling back along the period map the line bundle associated to $${{\,\mathrm{grif}\,}}(\mathbf {G}, \mu , r)$$ grif ( G , μ , r ) . The more general setting also gives rise to the Griffiths bundle in the analogous situation in characteristic p given by a scheme mapping to a stack of $$\mathbf {G}$$ G -Zips. When $$\mathbf {G}$$ G is F-simple, we show that, up to positive multiples, the Griffiths character $${{\,\mathrm{grif}\,}}(\mathbf {G},\mu ,r)$$ grif ( G , μ , r ) (and thus also the Griffiths line bundle) is essentially independent of r with central kernel, and up to some identifications is given explicitly by $$-\mu $$ - μ . As an application, we show that the Griffiths line bundle of a projective $${{\mathbf {G}{\text{- }}{} \mathtt{Zip}}}^{\mu }$$ G - Zip μ -scheme is nef.

Field-theoretic and linear-algebraic invariants

2015 ◽  
Author(s):  
Brian Conrad ◽  
Gopal Prasad
Keyword(s):  
Simple Group ◽  
Root System ◽  
Reductive Group ◽  
Simple Groups ◽  
Arbitrary Field ◽  
Isomorphism Classes ◽  
Algebraic Invariants ◽  
Characteristic 2 ◽  
Minimal Type

This chapter deals with field-theoretic and linear-algebraic invariants. It first presents a construction of non-standard pseudo-split absolutely pseudosimple k-groups with root system A1 over any imperfect field k of characteristic 2. It then considers an absolutely pseudo-simple group over a field k, along with a pseudo-split pseudo-reductive group over an arbitrary field k. It also establishes the equality over k of minimal fields of definition for projection onto maximal geometric adjoint semisimple quotients. This is followed by two examples that illustrate the root field in A1-cases. The chapter concludes with a discussion of a classification of the isomorphism classes of pseudo-split pseudo-simple groups G over an imperfect field k of characteristic p subject to the hypothesis that G is of minimal type. The associated irreducible root datum, which is sufficient to classify isomorphism classes in the semisimple case, is supplemented with additional field-theoretic and linear-algebraic data.

scholarly journals On the closure of the Hodge locus of positive period dimension

2021 ◽  
Author(s):  
B. Klingler ◽  
A. Otwinowska
Keyword(s):  
Moduli Space ◽  
Classical Result ◽  
Projective Variety ◽  
Abelian Varieties ◽  
Countable Union ◽  
Adjoint Group ◽  
Hodge Structures ◽  
Period Map ◽  
Tate Group ◽  
Irreducible Components

AbstractGiven $${{\mathbb {V}}}$$ V a polarizable variation of $${{\mathbb {Z}}}$$ Z -Hodge structures on a smooth connected complex quasi-projective variety S, the Hodge locus for $${{\mathbb {V}}}^\otimes $$ V ⊗ is the set of closed points s of S where the fiber $${{\mathbb {V}}}_s$$ V s has more Hodge tensors than the very general one. A classical result of Cattani, Deligne and Kaplan states that the Hodge locus for $${{\mathbb {V}}}^\otimes $$ V ⊗ is a countable union of closed irreducible algebraic subvarieties of S, called the special subvarieties of S for $${{\mathbb {V}}}$$ V . Under the assumption that the adjoint group of the generic Mumford–Tate group of $${{\mathbb {V}}}$$ V is simple we prove that the union of the special subvarieties for $${{\mathbb {V}}}$$ V whose image under the period map is not a point is either a closed algebraic subvariety of S or is Zariski-dense in S. This implies for instance the following typical intersection statement: given a Hodge-generic closed irreducible algebraic subvariety S of the moduli space $${{\mathcal {A}}}_g$$ A g of principally polarized Abelian varieties of dimension g, the union of the positive dimensional irreducible components of the intersection of S with the strict special subvarieties of $${{\mathcal {A}}}_g$$ A g is either a closed algebraic subvariety of S or is Zariski-dense in S.

Pro- Iwahori–Hecke algebras are Gorenstein

2013 ◽  
Vol 13 (4) ◽  
pp. 753-809 ◽  
Author(s):  
Rachel Ollivier ◽  
Peter Schneider
Keyword(s):  
Upper Bound ◽  
Reductive Group ◽  
Global Dimension ◽  
Arbitrary Field ◽  
Graded Rings ◽  
Injective Dimension ◽  
Locally Compact ◽  
Invariant Vectors ◽  
Residue Characteristic ◽  
Iwahori Subgroup

AbstractLet$\mathfrak{F}$be a locally compact nonarchimedean field with residue characteristic$p$, and let$\mathrm{G} $be the group of$\mathfrak{F}$-rational points of a connected split reductive group over$\mathfrak{F}$. For$k$an arbitrary field of any characteristic, we study the homological properties of the Iwahori–Hecke$k$-algebra${\mathrm{H} }^{\prime } $and of the pro-$p$Iwahori–Hecke$k$-algebra$\mathrm{H} $of$\mathrm{G} $. We prove that both of these algebras are Gorenstein rings with self-injective dimension bounded above by the rank of$\mathrm{G} $. If$\mathrm{G} $is semisimple, we also show that this upper bound is sharp, that both$\mathrm{H} $and${\mathrm{H} }^{\prime } $are Auslander–Gorenstein, and that there is a duality functor on the finite length modules of$\mathrm{H} $(respectively${\mathrm{H} }^{\prime } $). We obtain the analogous Gorenstein and Auslander–Gorenstein properties for the graded rings associated to$\mathrm{H} $and${\mathrm{H} }^{\prime } $.When$k$has characteristic$p$, we prove that in ‘most’ cases$\mathrm{H} $and${\mathrm{H} }^{\prime } $have infinite global dimension. In particular, we deduce that the category of smooth$k$-representations of$\mathrm{G} = {\mathrm{PGL} }_{2} ({ \mathbb{Q} }_{p} )$generated by their invariant vectors under the pro-$p$Iwahori subgroup has infinite global dimension (at least if$k$is algebraically closed).

Classification of all connected subgroup schemes of a reductive group containing a split maximal torus

2008 ◽  
Vol 3 (1) ◽  
pp. 103-122 ◽  
Author(s):  
Ekaterina Sopkina
Keyword(s):  
Root System ◽  
Maximal Torus ◽  
Reductive Group ◽  
Arbitrary Field ◽  
Connected Subgroup ◽  
Split Maximal Torus

AbstractThe main result of the paper is a classification of all connected subgroup schemes of a reductive group containing a split maximal torus, over an arbitrary field. The classification is expressed in terms of functions on the root system.

scholarly journals aCM vector bundles on projective surfaces of nonnegative Kodaira dimension

2021 ◽  
Author(s):  
Edoardo Ballico ◽  
Sukmoon Huh ◽  
Joan Pons-Llopis
Keyword(s):  
Positive Integer ◽  
Line Bundle ◽  
Vector Bundles ◽  
Ample Line Bundle ◽  
General Setting ◽  
Kodaira Dimension ◽  
Line Bundles ◽  
Wide Range ◽  
Wild Representation Type ◽  
Projective Surfaces

In this paper, we contribute to the construction of families of arithmetically Cohen–Macaulay (aCM) indecomposable vector bundles on a wide range of polarized surfaces [Formula: see text] for [Formula: see text] an ample line bundle. In many cases, we show that for every positive integer [Formula: see text] there exists a family of indecomposable aCM vector bundles of rank [Formula: see text], depending roughly on [Formula: see text] parameters, and in particular they are of wild representation type. We also introduce a general setting to study the complexity of a polarized variety [Formula: see text] with respect to its category of aCM vector bundles. In many cases we construct indecomposable vector bundles on [Formula: see text] which are aCM for all ample line bundles on [Formula: see text].

Short Generating Functions for some Semigroup Algebras

10.37236/1729 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Graham Denham
Keyword(s):  
Rational Function ◽  
Generating Functions ◽  
Hilbert Series ◽  
Simple Expression ◽  
Arbitrary Field ◽  
Graded Algebra ◽  
Semigroup Algebras ◽  
Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

scholarly journals Directed unions of local monoidal transforms and GCD domains

2019 ◽  
Vol 19 (04) ◽  
pp. 2050061
Author(s):  
Lorenzo Guerrieri
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Maximal Ideal ◽  
General Setting ◽  
Regular Local Ring ◽  
Dimension Formula

Let [Formula: see text] be a regular local ring of dimension [Formula: see text]. A local monoidal transform of [Formula: see text] is a ring of the form [Formula: see text], where [Formula: see text] is a regular parameter, [Formula: see text] is a regular prime ideal of [Formula: see text] and [Formula: see text] is a maximal ideal of [Formula: see text] lying over [Formula: see text] In this paper, we study some features of the rings [Formula: see text] obtained as infinite directed union of iterated local monoidal transforms of [Formula: see text]. In order to study when these rings are GCD domains, we also provide results in the more general setting of directed unions of GCD domains.

scholarly journals Explicit soft supersymmetry breaking in the heterotic M-theory B − L MSSM

2021 ◽  
Vol 2021 (8) ◽  
Author(s):  
Anthony Ashmore ◽  
Sebastian Dumitru ◽  
Burt A. Ovrut
Keyword(s):  
Line Bundle ◽  
Supersymmetry Breaking ◽  
Moduli Space ◽  
Initial Conditions ◽  
Low Energy ◽  
Strongly Coupled ◽  
Hidden Sector ◽  
Effective Lagrangian ◽  
Soft Supersymmetry Breaking ◽  
M Theory

Abstract The strongly coupled heterotic M-theory vacuum for both the observable and hidden sectors of the B − L MSSM theory is reviewed, including a discussion of the “bundle” constraints that both the observable sector SU(4) vector bundle and the hidden sector bundle induced from a single line bundle must satisfy. Gaugino condensation is then introduced within this context, and the hidden sector bundles that exhibit gaugino condensation are presented. The condensation scale is computed, singling out one line bundle whose associated condensation scale is low enough to be compatible with the energy scales available at the LHC. The corresponding region of Kähler moduli space where all bundle constraints are satisfied is presented. The generic form of the moduli dependent F-terms due to a gaugino superpotential — which spontaneously break N = 1 supersymmetry in this sector — is presented and then given explicitly for the unique line bundle associated with the low condensation scale. The moduli-dependent coefficients for each of the gaugino and scalar field soft supersymmetry breaking terms are computed leading to a low-energy effective Lagrangian for the observable sector matter fields. We then show that at a large number of points in Kähler moduli space that satisfy all “bundle” constraints, these coefficients are initial conditions for the renormalization group equations which, at low energy, lead to completely realistic physics satisfying all phenomenological constraints. Finally, we show that a substantial number of these initial points also satisfy a final constraint arising from the quadratic Higgs-Higgs conjugate soft supersymmetry breaking term.

scholarly journals Irreducible modules with highest weight vectors over modular Witt and special Lie superalgebras

2019 ◽  
Vol 17 (1) ◽  
pp. 1381-1391
Author(s):  
Keli Zheng ◽  
Yongzheng Zhang
Keyword(s):  
Lie Superalgebras ◽  
Arbitrary Field ◽  
Special Linear ◽  
Highest Weight ◽  
Irreducible Modules

Abstract Let 𝔽 be an arbitrary field of characteristic p > 2. In this paper we study irreducible modules with highest weight vectors over Witt and special Lie superalgebras of 𝔽. The same irreducible modules of general and special linear Lie superalgebras, which are the 0-th part of Witt and special Lie superalgebras in certain ℤ-grading, are also considered. Then we establish a certain connection called a P-expansion between these modules.

Sign in / Sign up

Export Citation Format

Share Document