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Tian’s criterion for K-stability states that a Fano variety of dimension n whose alpha invariant is greater than
${n}{/(n+1)}$
is K-stable. We show that this criterion is sharp by constructing n-dimensional singular Fano varieties with alpha invariants
${n}{/(n+1)}$
that are not K-polystable for sufficiently large n. We also construct K-unstable Fano varieties with alpha invariants
${(n-1)}{/n}$
.
Rapoport–Zink spaces are deformation spaces for
$p$
-divisible groups with additional structure. At infinite level, they become preperfectoid spaces. Let
${{\mathscr M}}_{\infty }$
be an infinite-level Rapoport–Zink space of EL type, and let
${{\mathscr M}}_{\infty }^{\circ }$
be one connected component of its geometric fiber. We show that
${{\mathscr M}}_{\infty }^{\circ }$
contains a dense open subset which is cohomologically smooth in the sense of Scholze. This is the locus of
$p$
-divisible groups which do not have any extra endomorphisms. As a corollary, we find that the cohomologically smooth locus in the infinite-level modular curve
$X(p^{\infty })^{\circ }$
is exactly the locus of elliptic curves
$E$
with supersingular reduction, such that the formal group of
$E$
has no extra endomorphisms.
Let A be the product of an abelian variety and a torus over a number field K, and let
$$m \ge 2$$
be a square-free integer. If
$\alpha \in A(K)$
is a point of infinite order, we consider the set of primes
$\mathfrak p$
of K such that the reduction
$(\alpha \bmod \mathfrak p)$
is well defined and has order coprime to m. This set admits a natural density, which we are able to express as a finite sum of products of
$\ell$
-adic integrals, where
$\ell$
varies in the set of prime divisors of m. We deduce that the density is a rational number, whose denominator is bounded (up to powers of m) in a very strong sense. This extends the results of the paper Reductions of points on algebraic groups by Davide Lombardo and the second author, where the case m prime is established.
Bernstein, Frenkel, and Khovanov have constructed a categorification of tensor products of the standard representation of
$\mathfrak {sl}_2$
, where they use singular blocks of category
$\mathcal {O}$
for
$\mathfrak {sl}_n$
and translation functors. Here we construct a positive characteristic analogue using blocks of representations of
$\mathfrak {s}\mathfrak {l}_n$
over a field
$\mathbf {k}$
of characteristic p with zero Frobenius character, and singular Harish-Chandra character. We show that the aforementioned categorification admits a Koszul graded lift, which is equivalent to a geometric categorification constructed by Cautis, Kamnitzer, and Licata using coherent sheaves on cotangent bundles to Grassmanians. In particular, the latter admits an abelian refinement. With respect to this abelian refinement, the stratified Mukai flop induces a perverse equivalence on the derived categories for complementary Grassmanians. This is part of a larger project to give a combinatorial approach to Lusztig’s conjectures for representations of Lie algebras in positive characteristic.
We study exceptional Jordan algebras and related exceptional group schemes over commutative rings from a geometric point of view, using appropriate torsors to parametrize and explain classical and new constructions, and proving that over rings, they give rise to nonisomorphic structures.
We begin by showing that isotopes of Albert algebras are obtained as twists by a certain
$\mathrm F_4$
-torsor with total space a group of type
$\mathrm E_6$
and, using this, that Albert algebras over rings in general admit nonisomorphic isotopes even in the split case, as opposed to the situation over fields. We then consider certain
$\mathrm D_4$
-torsors constructed from reduced Albert algebras, and show how these give rise to a class of generalised reduced Albert algebras constructed from compositions of quadratic forms. Showing that this torsor is nontrivial, we conclude that the Albert algebra does not uniquely determine the underlying composition, even in the split case. In a similar vein, we show that a given reduced Albert algebra can admit two coordinate algebras which are nonisomorphic and have nonisometric quadratic forms, contrary, in a strong sense, to the case over fields, established by Albert and Jacobson.
We consider the Newton stratification on Iwahori-double cosets in the loop group of a reductive group. We describe a group-theoretic condition on the generic Newton point, called cordiality, under which the Newton poset (that is, the index set for non-empty Newton strata) is saturated and Grothendieck’s conjecture on closures of the Newton strata holds. Finally, we give several large classes of Iwahori-double cosets for which this condition is satisfied by studying certain paths in the associated quantum Bruhat graph.
Let G be a reductive algebraic group—possibly non-connected—over a field k, and let H be a subgroup of G. If
$G= {GL }_n$
, then there is a degeneration process for obtaining from H a completely reducible subgroup
$H'$
of G; one takes a limit of H along a cocharacter of G in an appropriate sense. We generalise this idea to arbitrary reductive G using the notion of G-complete reducibility and results from geometric invariant theory over non-algebraically closed fields due to the authors and Herpel. Our construction produces a G-completely reducible subgroup
$H'$
of G, unique up to
$G(k)$
-conjugacy, which we call a k-semisimplification of H. This gives a single unifying construction that extends various special cases in the literature (in particular, it agrees with the usual notion for
$G= GL _n$
and with Serre’s ‘G-analogue’ of semisimplification for subgroups of
$G(k)$
from [19]). We also show that under some extra hypotheses, one can pick
$H'$
in a more canonical way using the Tits Centre Conjecture for spherical buildings and/or the theory of optimal destabilising cocharacters introduced by Hesselink, Kempf, and Rousseau.
We study a relative variant of Serre’s notion of
$G$
-complete reducibility for a reductive algebraic group
$G$
. We let
$K$
be a reductive subgroup of
$G$
, and consider subgroups of
$G$
that normalize the identity component
$K^{\circ }$
. We show that such a subgroup is relatively
$G$
-completely reducible with respect to
$K$
if and only if its image in the automorphism group of
$K^{\circ }$
is completely reducible. This allows us to generalize a number of fundamental results from the absolute to the relative setting. We also derive analogous results for Lie subalgebras of the Lie algebra of
$G$
, as well as ‘rational’ versions over nonalgebraically closed fields.
Let
$A$
be the product of an abelian variety and a torus defined over a number field
$K$
. Fix some prime number
$\ell$
. If
$\unicode[STIX]{x1D6FC}\in A(K)$
is a point of infinite order, we consider the set of primes
$\mathfrak{p}$
of
$K$
such that the reduction
$(\unicode[STIX]{x1D6FC}\hspace{0.2em}{\rm mod}\hspace{0.2em}\mathfrak{p})$
is well-defined and has order coprime to
$\ell$
. This set admits a natural density. By refining the method of Jones and Rouse [Galois theory of iterated endomorphisms, Proc. Lond. Math. Soc. (3)100(3) (2010), 763–794. Appendix A by Jeffrey D. Achter], we can express the density as an
$\ell$
-adic integral without requiring any assumption. We also prove that the density is always a rational number whose denominator (up to powers of
$\ell$
) is uniformly bounded in a very strong sense. For elliptic curves, we describe a strategy for computing the density which covers every possible case.
Nous montrons, pour une grande famille de propriétés des espaces homogènes, qu’une telle propriété vaut pour tout espace homogène d’un groupe linéaire connexe dès qu’elle vaut pour les espaces homogènes de
$\text{SL}_{n}$
à stabilisateur fini. Nous réduisons notamment à ce cas particulier la vérification d’une importante conjecture de Colliot-Thélène sur l’obstruction de Brauer–Manin au principe de Hasse et à l’approximation faible. Des travaux récents de Harpaz et Wittenberg montrent que le résultat principal s’applique également à la conjecture analogue (dite conjecture (E)) pour les zéro-cycles.
In this article we construct a p-adic three-dimensional eigenvariety for the group
$U$
(2,1)(
$E$
), where
$E$
is a quadratic imaginary field and
$p$
is inert in
$E$
. The eigenvariety parametrizes Hecke eigensystems on the space of overconvergent, locally analytic, cuspidal Picard modular forms of finite slope. The method generalized the one developed in Andreatta, Iovita and Stevens [
$p$
-adic families of Siegel modular cuspforms Ann. of Math. (2) 181, (2015), 623–697] by interpolating the coherent automorphic sheaves when the ordinary locus is empty. As an application of this construction, we reprove a particular case of the Bloch–Kato conjecture for some Galois characters of
$E$
, extending the results of Bellaiche and Chenevier to the case of a positive sign.
In this paper, we prove that the set of all
$F$
-pure thresholds on a fixed germ of a strongly
$F$
-regular pair satisfies the ascending chain condition. As a corollary, we verify the ascending chain condition for the set of all
$F$
-pure thresholds on smooth varieties or, more generally, on varieties with tame quotient singularities, which is an affirmative answer to a conjecture given by Blickle, Mustaţǎ and Smith.
We develop the analog of crystalline Dieudonné theory for
$p$
-divisible groups in the arithmetic of function fields. In our theory
$p$
-divisible groups are replaced by divisible local Anderson modules, and Dieudonné modules are replaced by local shtukas. We show that the categories of divisible local Anderson modules and of effective local shtukas are anti-equivalent over arbitrary base schemes. We also clarify their relation with formal Lie groups and with global objects like Drinfeld modules, Anderson’s abelian
$t$
-modules and
$t$
-motives, and Drinfeld shtukas. Moreover, we discuss the existence of a Verschiebung map and apply it to deformations of local shtukas and divisible local Anderson modules. As a tool we use Faltings’s and Abrashkin’s theories of strict modules, which we review briefly.
A polarized variety is K-stable if, for any test configuration, the Donaldson–Futaki invariant is positive. In this paper, inspired by classical geometric invariant theory, we describe the space of test configurations as a limit of a direct system of Tits buildings. We show that the Donaldson–Futaki invariant, conveniently normalized, is a continuous function on this space. We also introduce a pseudo-metric on the space of test configurations. Recall that K-stability can be enhanced by requiring that the Donaldson–Futaki invariant is positive on any admissible filtration of the co-ordinate ring. We show that admissible filtrations give rise to Cauchy sequences of test configurations with respect to the above mentioned pseudo-metric.
We exhibit invariants of smooth projective algebraic varieties with integer values, whose nonvanishing modulo
$p$
prevents the existence of an action without fixed points of certain finite
$p$
-groups. The case of base fields of characteristic
$p$
is included. Counterexamples are systematically provided to test the sharpness of our results.
We describe all degenerations of three-dimensional anticommutative algebras
$\mathfrak{A}\mathfrak{c}\mathfrak{o}\mathfrak{m}_{3}$
and of three-dimensional Leibniz algebras
$\mathfrak{L}\mathfrak{e}\mathfrak{i}\mathfrak{b}_{3}$
over
$\mathbb{C}$
. In particular, we describe all irreducible components and rigid algebras in the corresponding varieties.
According to a well-known theorem of Serre and Tate, the infinitesimal deformation theory of an abelian variety in positive characteristic is equivalent to the infinitesimal deformation theory of its Barsotti–Tate group. We extend this result to 1-motives.
Let
$X:=\mathbb{A}_{R}^{n}$
be the
$n$
-dimensional affine space over a discrete valuation ring
$R$
with fraction field
$K$
. We prove that any pointed torsor
$Y$
over
$\mathbb{A}_{K}^{n}$
under the action of an affine finite-type group scheme can be extended to a torsor over
$\mathbb{A}_{R}^{n}$
possibly after pulling
$Y$
back over an automorphism of
$\mathbb{A}_{K}^{n}$
. The proof is effective. Other cases, including
$X=\unicode[STIX]{x1D6FC}_{p,R}$
, are also discussed.
We systematically produce algebraic varieties with torus action by constructing them as suitably embedded subvarieties of toric varieties. The resulting varieties admit an explicit treatment in terms of toric geometry and graded ring theory. Our approach extends existing constructions of rational varieties with torus action of complexity one and delivers all Mori dream spaces with torus action. We exhibit the example class of ‘general arrangement varieties’ and obtain classification results in the case of complexity two and Picard number at most two, extending former work in complexity one.
Let U be the unipotent radical of a Borel subgroup of a connected reductive algebraic group G, which is defined over an algebraically closed field k. In this paper, we extend work by Goodwin and Röhrle concerning the commuting variety of Lie(U) for Char(k) = 0 to fields whose characteristic is good for G.