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We generalize the works of Pappas–Rapoport–Zhu on twisted affine Grassmannians to the wildly ramified case under mild assumptions. This rests on a construction of certain smooth affine
$\mathbb {Z}[t]$
-groups with connected fibers of parahoric type, motivated by previous work of Tits. The resulting
$\mathbb {F}_p(t)$
-groups are pseudo-reductive and sometimes non-standard in the sense of Conrad–Gabber–Prasad, and their
$\mathbb {F}_p [\hspace {-0,5mm}[ {t} ]\hspace {-0,5mm}] $
-models are parahoric in a generalized sense. We study their affine Grassmannians, proving normality of Schubert varieties and Zhu’s coherence theorem.
Let V be a smooth quasi-projective complex surface such that the first three logarithmic plurigenera
$\overline P_1(V)$
,
$\overline P_2(V)$
and
$\overline P_3(V)$
are equal to 1 and the logarithmic irregularity
$\overline q(V)$
is equal to
$2$
. We prove that the quasi-Albanese morphism
$a_V\colon V\to A(V)$
is birational and there exists a finite set S such that
$a_V$
is proper over
$A(V)\setminus S$
, thus giving a sharp effective version of a classical result of Iitaka [12].
We define, for each quasisyntomic ring R (in the sense of Bhatt et al., Publ. Math. IHES129 (2019), 199–310), a category
$\mathrm {DM}^{\mathrm {adm}}(R)$
of admissible prismatic Dieudonné crystals over R and a functor from p-divisible groups over R to
$\mathrm {DM}^{\mathrm {adm}}(R)$
. We prove that this functor is an antiequivalence. Our main cohomological tool is the prismatic formalism recently developed by Bhatt and Scholze.
An affine variety with an action of a semisimple group G is called “small” if every nontrivial G-orbit in X is isomorphic to the orbit of a highest weight vector. Such a variety X carries a canonical action of the multiplicative group
${\mathbb {K}^{*}}$
commuting with the G-action. We show that X is determined by the
${\mathbb {K}^{*}}$
-variety
$X^U$
of fixed points under a maximal unipotent subgroup
$U \subset G$
. Moreover, if X is smooth, then X is a G-vector bundle over the algebraic quotient
$X /\!\!/ G$
.
If G is of type
${\mathsf {A}_n}$
(
$n\geq 2$
),
${\mathsf {C}_{n}}$
,
${\mathsf {E}_{6}}$
,
${\mathsf {E}_{7}}$
, or
${\mathsf {E}_{8}}$
, we show that all affine G-varieties up to a certain dimension are small. As a consequence, we have the following result. If
$n \geq 5$
, every smooth affine
$\operatorname {\mathrm {SL}}_n$
-variety of dimension
$< 2n-2$
is an
$\operatorname {\mathrm {SL}}_n$
-vector bundle over the smooth quotient
$X /\!\!/ \operatorname {\mathrm {SL}}_n$
, with fiber isomorphic to the natural representation or its dual.
We show that the only finite quasi-simple non-abelian groups that can faithfully act on rationally connected threefolds are the following groups:
${\mathfrak{A}}_5$
,
${\text{PSL}}_2(\textbf{F}_7)$
,
${\mathfrak{A}}_6$
,
${\text{SL}}_2(\textbf{F}_8)$
,
${\mathfrak{A}}_7$
,
${\text{PSp}}_4(\textbf{F}_3)$
,
${\text{SL}}_2(\textbf{F}_{7})$
,
$2.{\mathfrak{A}}_5$
,
$2.{\mathfrak{A}}_6$
,
$3.{\mathfrak{A}}_6$
or
$6.{\mathfrak{A}}_6$
. All of these groups with a possible exception of
$2.{\mathfrak{A}}_6$
and
$6.{\mathfrak{A}}_6$
indeed act on some rationally connected threefolds.
In their renowned paper (2011, Inventiones Mathematicae 184, 591–627), I. Vollaard and T. Wedhorn defined a stratification on the special fiber of the unitary unramified PEL Rapoport–Zink space with signature $(1,n-1)$. They constructed an isomorphism between the closure of a stratum, called a closed Bruhat–Tits stratum, and a Deligne–Lusztig variety which is not of classical type. In this paper, we describe the $\ell $-adic cohomology groups over $\overline {{\mathbb Q}_{\ell }}$ of these Deligne–Lusztig varieties, where $\ell \not = p$. The computations involve the spectral sequence associated with the Ekedahl–Oort stratification of a closed Bruhat–Tits stratum, which translates into a stratification by Coxeter varieties whose cohomology is known. Eventually, we find out that the irreducible representations of the finite unitary group which appear inside the cohomology contribute to only two different unipotent Harish-Chandra series, one of them belonging to the principal series.
We develop the formalism of universal torsors in equivariant birational geometry and apply it to produce new examples of nonbirational but stably birational actions of finite groups.
We compute the number of points over finite fields of the character stack associated to a compact surface group and a reductive group with connected centre. We find that the answer is a polynomial on residue classes (PORC). The key ingredients in the proof are Lusztig’s Jordan decomposition of complex characters of finite reductive groups and Deriziotis’s results on their genus numbers. As a consequence of our main theorem, we obtain an expression for the E-polynomial of the character stack.
In characteristic
$0$
, symplectic automorphisms of K3 surfaces (i.e., automorphisms preserving the global
$2$
-form) and non-symplectic ones behave differently. In this paper, we consider the actions of the group schemes
$\mu _{n}$
on K3 surfaces (possibly with rational double point [RDP] singularities) in characteristic p, where n may be divisible by p. We introduce the notion of symplecticness of such actions, and we show that symplectic
$\mu _{n}$
-actions have similar properties, such as possible orders, fixed loci, and quotients, to symplectic automorphisms of order n in characteristic
$0$
. We also study local
$\mu _n$
-actions on RDPs.
We show that the image of a subshift X under various injective morphisms of symbolic algebraic varieties over monoid universes with algebraic variety alphabets is a subshift of finite type, respectively a sofic subshift, if and only if so is X. Similarly, let G be a countable monoid and let A, B be Artinian modules over a ring. We prove that for every closed subshift submodule
$\Sigma \subset A^G$
and every injective G-equivariant uniformly continuous module homomorphism
$\tau \colon \! \Sigma \to B^G$
, a subshift
$\Delta \subset \Sigma $
is of finite type, respectively sofic, if and only if so is the image
$\tau (\Delta )$
. Generalizations for admissible group cellular automata over admissible Artinian group structure alphabets are also obtained.
In this paper, we study sample size thresholds for maximum likelihood estimation for tensor normal models. Given the model parameters and the number of samples, we determine whether, almost surely, (1) the likelihood function is bounded from above, (2) maximum likelihood estimates (MLEs) exist, and (3) MLEs exist uniquely. We obtain a complete answer for both real and complex models. One consequence of our results is that almost sure boundedness of the log-likelihood function guarantees almost sure existence of an MLE. Our techniques are based on invariant theory and castling transforms.
Let $p$ be a rational prime, let $F$ denote a finite, unramified extension of ${{\mathbb {Q}}}_p$, let $K$ be the maximal unramified extension of ${{\mathbb {Q}}}_p$, ${{\overline {K}}}$ some fixed algebraic closure of $K$, and ${{\mathbb {C}}}_p$ be the completion of ${{\overline {K}}}$. Let $G_F$ be the absolute Galois group of $F$. Let $A$ be an abelian variety defined over $F$, with good reduction. Classically, the Fontaine integral was seen as a Hodge–Tate comparison morphism, i.e. as a map $\varphi _{A} \otimes 1_{{{\mathbb {C}}}_p}\colon T_p(A)\otimes _{{{\mathbb {Z}}}_p}{{\mathbb {C}}}_p\to \operatorname {Lie}(A)(F)\otimes _F{{\mathbb {C}}}_p(1)$, and as such it is surjective and has a large kernel. This paper starts with the observation that if we do not tensor $T_p(A)$ with ${{\mathbb {C}}}_p$, then the Fontaine integral is often injective. In particular, it is proved that if $T_p(A)^{G_K} = 0$, then $\varphi _A$ is injective. As an application, we extend the Fontaine integral to a perfectoid like universal cover of $A$ and show that if $T_p(A)^{G_K} = 0$, then $A(\overline {K})$ has a type of $p$-adic uniformization, which resembles the classical complex uniformization.
The Grothendieck–Serre conjecture predicts that every generically trivial torsor under a reductive group scheme G over a regular local ring R is trivial. We settle it in the case when G is quasi-split and R is unramified. Some of the techniques that allow us to overcome obstacles that have so far kept the mixed characteristic case out of reach include a version of Noether normalization over discrete valuation rings, as well as a suitable presentation lemma for smooth relative curves in mixed characteristic that facilitates passage to the relative affine line via excision and patching.
We study model theory of fields with actions of a fixed finite group scheme. We prove the existence and simplicity of a model companion of the theory of such actions, which generalizes our previous results about truncated iterative Hasse–Schmidt derivations [13] and about Galois actions [14]. As an application of our methods, we obtain a new model complete theory of actions of a finite group on fields of finite imperfection degree.
We study reductive subgroups H of a reductive linear algebraic group G – possibly nonconnected – such that H contains a regular unipotent element of G. We show that under suitable hypotheses, such subgroups are G-irreducible in the sense of Serre. This generalises results of Malle, Testerman and Zalesski. We obtain analogous results for Lie algebras and for finite groups of Lie type. Our proofs are short, conceptual and uniform.
In this article we study integral models of Shimura varieties, called Pappas–Rapoport splitting model, for ramified P.E.L. Shimira data. We study the special fiber and some stratification of these models, in particular we show that these are smooth and the Rapoport locus and the
$\mu $
-ordinary locus are dense, under some condition on the ramification.
We prove that if G is a finite flat group scheme of p-power rank over a perfect field of characteristic p, then the second crystalline cohomology of its classifying stack $H^2_{\text {crys}}(BG)$ recovers the Dieudonné module of G. We also provide a calculation of the crystalline cohomology of the classifying stack of an abelian variety. We use this to prove that the crystalline cohomology of the classifying stack of a p-divisible group is a symmetric algebra (in degree $2$) on its Dieudonné module. We also prove mixed-characteristic analogues of some of these results using prismatic cohomology.
For $G = \mathrm {GL}_2, \mathrm {SL}_2, \mathrm {PGL}_2$ we compute the intersection E-polynomials and the intersection Poincaré polynomials of the G-character variety of a compact Riemann surface C and of the moduli space of G-Higgs bundles on C of degree zero. We derive several results concerning the P=W conjectures for these singular moduli spaces.
We consider G, a linear algebraic group defined over
$\Bbbk $
, an algebraically closed field (ACF). By considering
$\Bbbk $
as an embedded residue field of an algebraically closed valued field K, we can associate to it a compact G-space
$S^\mu _G(\Bbbk )$
consisting of
$\mu $
-types on G. We show that for each
$p_\mu \in S^\mu _G(\Bbbk )$
,
$\mathrm {Stab}^\mu (p)=\mathrm {Stab}\left (p_\mu \right )$
is a solvable infinite algebraic group when
$p_\mu $
is centered at infinity and residually algebraic. Moreover, we give a description of the dimension of
$\mathrm {Stab}\left (p_\mu \right )$
in terms of the dimension of p.