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In this note, we prove that the moduli stack of vector bundles on a curve with a fixed determinant is
${\mathbb A}^1$
-connected. We obtain this result by classifying vector bundles on a curve up to
${\mathbb A}^1$
-concordance. Consequently, we classify
${\mathbb P}^n$
-bundles on a curve up to
${\mathbb A}^1$
-weak equivalence, extending a result in [3] of Asok-Morel. We also give an explicit example of a variety which is
${\mathbb A}^1$
-h-cobordant to a projective bundle over
${\mathbb P}^2$
but does not have the structure of a projective bundle over
${\mathbb P}^2$
, thus answering a question of Asok-Kebekus-Wendt [2].
We prove some
$\ell $
-independence results on local constancy of étale cohomology of rigid analytic varieties. As a result, we show that a closed subscheme of a proper scheme over an algebraically closed complete non-archimedean field has a small open neighbourhood in the analytic topology such that, for every prime number
$\ell $
different from the residue characteristic, the closed subscheme and the open neighbourhood have the same étale cohomology with
${\mathbb Z}/\ell {\mathbb Z}$
-coefficients. The existence of such an open neighbourhood for each
$\ell $
was proved by Huber. A key ingredient in the proof is a uniform refinement of a theorem of Orgogozo on the compatibility of the nearby cycles over general bases with base change.
Jannsen asked whether the rational cycle class map in continuous
$\ell $
-adic cohomology is injective, in every codimension for all smooth projective varieties over a field of finite type over the prime field. As recently pointed out by Schreieder, the integral version of Jannsen’s question is also of interest. We exhibit several examples showing that the answer to the integral version is negative in general. Our examples also have consequences for the coniveau filtration on Chow groups and the transcendental Abel-Jacobi map constructed by Schreieder.
We connect two developments that aim to extend Voevodsky's theory of motives over a field in such a way as to encompass non-$\mathbf {A}^1$-invariant phenomena. One is theory of reciprocity sheaves introduced by Kahn, Saito and Yamazaki. The other is theory of the triangulated category $\operatorname {\mathbf {logDM}}^{{\operatorname {eff}}}$ of logarithmic motives launched by Binda, Park and Østvær. We prove that the Nisnevich cohomology of reciprocity sheaves is representable in $\operatorname {\mathbf {logDM}}^{{\operatorname {eff}}}$.
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.
The primary goal of this paper is to identify syntomic complexes with the p-adic étale Tate twists of Geisser–Sato–Schneider on regular p-torsion-free schemes. Our methods apply naturally to a broader class of schemes that we call ‘F-smooth’. The F-smoothness of regular schemes leads to new results on the absolute prismatic cohomology of regular schemes.
We prove that every element of order 2 in the Brauer group of a complex Kummer surface X descends to an Enriques quotient of X. In generic cases, this gives a bijection between the set ${\mathcal Enr}(X)$ of Enriques quotients of X up to isomorphism and the set of Brauer classes of X of order 2. For some K3 surfaces of Picard rank $20,$ we prove that the fibers of ${\mathcal Enr}(X)\to \mathrm {{Br}}(X)[2]$ above the nonzero points have the same cardinality.
Utilizing ultraproducts, Schoutens constructed a big Cohen–Macaulay (BCM) algebra
$\mathcal {B}(R)$
over a local domain R essentially of finite type over
$\mathbb {C}$
. We show that if R is normal and
$\Delta $
is an effective
$\mathbb {Q}$
-Weil divisor on
$\operatorname {Spec} R$
such that
$K_R+\Delta $
is
$\mathbb {Q}$
-Cartier, then the BCM test ideal
$\tau _{\widehat {\mathcal {B}(R)}}(\widehat {R},\widehat {\Delta })$
of
$(\widehat {R},\widehat {\Delta })$
with respect to
$\widehat {\mathcal {B}(R)}$
coincides with the multiplier ideal
$\mathcal {J}(\widehat {R},\widehat {\Delta })$
of
$(\widehat {R},\widehat {\Delta })$
, where
$\widehat {R}$
and
$\widehat {\mathcal {B}(R)}$
are the
$\mathfrak {m}$
-adic completions of R and
$\mathcal {B}(R)$
, respectively, and
$\widehat {\Delta }$
is the flat pullback of
$\Delta $
by the canonical morphism
$\operatorname {Spec} \widehat {R}\to \operatorname {Spec} R$
. As an application, we obtain a result on the behavior of multiplier ideals under pure ring extensions.
O-minimal geometry generalizes both semialgebraic and subanalytic geometries, and has been very successful in solving special cases of some problems in arithmetic geometry, such as André–Oort conjecture. Among the many tools developed in an o-minimal setting are cohomology theories for abstract-definable continuous manifolds such as singular cohomology, sheaf cohomology and Čech cohomology, which have been used for instance to prove Pillay’s conjecture concerning definably compact groups. In the present thesis we elaborate an o-minimal de Rham cohomology theory for abstract-definable
$C^{\infty }$
manifolds in an o-minimal expansion of the real field which admits smooth cell decomposition and defines the exponential function. We can specify the o-minimal cohomology groups and attain some properties such as the existence of Mayer–Vietoris sequence and the invariance under abstract-definable
$C^{\infty }$
diffeomorphisms. However, in order to obtain the invariance of our o-minimal cohomology under abstract-definable homotopy we must work in a tame context that defines sufficiently many primitives and assume the validity of a statement related to Bröcker’s question.
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 study non-abelian versions of the Mellin transformations, originally introduced by Gabber-Loeser on complex affine tori. Our main result is a generalisation to the non-abelian context and with arbitrary coefficients of the t-exactness of Gabber-Loeser’s Mellin transformation. As an intermediate step, we obtain vanishing results for the Sabbah specialisation functors. Our main application is to construct new examples of duality spaces in the sense of Bieri-Eckmann, generalising results of Denham-Suciu.
We extend Poincaré duality in étale cohomology from smooth schemes to regular ones. This is achieved via a formalism of trace maps for local complete intersection morphisms.
We study the Hodge filtration on the local cohomology sheaves of a smooth complex algebraic variety along a closed subscheme Z in terms of log resolutions and derive applications regarding the local cohomological dimension, the Du Bois complex, local vanishing and reflexive differentials associated to Z.
We define cohomological complexes of locally compact abelian groups associated with varieties over p-adic fields and prove a duality theorem under some assumption. Our duality takes the form of Pontryagin duality between locally compact motivic cohomology groups.
The goal of this paper is to raise the possibility that there exists a meaningful theory of ‘motives’ associated with certain hypergeometric integrals, viewed as functions of their parameters. It goes beyond the classical theory of motives, but should be compatible with it. Such a theory would explain a recent and surprising conjecture arising in the context of scattering amplitudes for a motivic Galois group action on Gauss’s
${}_2F_1$
hypergeometric function, which we prove in this paper by direct means. More generally, we consider Lauricella hypergeometric functions and show, on the one hand, how the coefficients in their Taylor expansions can be promoted, via the theory of motivic fundamental groups, to motivic multiple polylogarithms. The latter are periods of ordinary motives and admit an action of the usual motivic Galois group, which we call the local action. On the other hand, we define lifts of the full Lauricella functions as matrix coefficients in a Tannakian category of twisted cohomology, which inherit an action of the corresponding Tannaka group. We call this the global action. We prove that these two actions, local and global, are compatible with each other, even though they are defined in completely different ways. The main technical tool is to prove that metabelian quotients of generalised Drinfeld associators on the punctured Riemann sphere are hypergeometric functions. We also study single-valued versions of these hypergeometric functions, which may be of independent interest.
For a smooth rigid space X over a perfectoid field extension K of
$\mathbb {Q}_p$
, we investigate how the v-Picard group of the associated diamond
$X^{\diamondsuit }$
differs from the analytic Picard group of X. To this end, we construct a left-exact ‘Hodge–Tate logarithm’ sequence
We deduce some analyticity criteria which have applications to p-adic modular forms. For algebraically closed K, we show that the sequence is also right-exact if X is proper or one-dimensional. In contrast, we show that, for the affine space
$\mathbb {A}^n$
, the image of the Hodge–Tate logarithm consists precisely of the closed differentials. It follows that, up to a splitting, v-line bundles may be interpreted as Higgs bundles. For proper X, we use this to construct the p-adic Simpson correspondence of rank one.
In this paper, we show the existence of an action of Chow correspondences on the cohomology of reciprocity sheaves. In order to do so, we prove a number of structural results, such as a projective bundle formula, a blow-up formula, a Gysin sequence and the existence of proper pushforward. In this way, we recover and generalise analogous statements for the cohomology of Hodge sheaves and Hodge-Witt sheaves.
We give several applications of the general theory to problems which have been classically studied. Among these applications, we construct new birational invariants of smooth projective varieties and obstructions to the existence of zero cycles of degree 1 from the cohomology of reciprocity sheaves.
We offer a systematic study of rigid analytic motives over general rigid analytic spaces, and we develop their six-functor formalism. A key ingredient is an extended proper base change theorem that we are able to justify by reducing to the case of algebraic motives. In fact, more generally, we develop a powerful technique for reducing questions about rigid analytic motives to questions about algebraic motives, which is likely to be useful in other contexts as well. We pay special attention to establishing our results without noetherianity assumptions on rigid analytic spaces. This is indeed possible using Raynaud’s approach to rigid analytic geometry.
from the de Jong fundamental group of the rigid generic fiber to the Bhatt–Scholze pro-étale fundamental group of the special fiber. The construction relies on an interplay between admissible blowups of $\mathfrak {X}$ and normalizations of the irreducible components of $\mathfrak {X}_k$, and employs the Berthelot tubes of these irreducible components in an essential way. Using related techniques, we show that under certain smoothness and semistability assumptions, covering spaces in the sense of de Jong of a smooth rigid space which are tame satisfy étale descent.
Recent works at the interface of algebraic combinatorics, algebraic geometry, number theory and topology have provided new integer-valued invariants on integer partitions. It is natural to consider the distribution of partitions when sorted by these invariants in congruence classes. We consider the prominent situations that arise from extensions of the Nekrasov–Okounkov hook product formula and from Betti numbers of various Hilbert schemes of n points on
${\mathbb {C}}^2$
. For the Hilbert schemes, we prove that homology is equidistributed as
$n\to \infty $
. For t-hooks, we prove distributions that are often not equidistributed. The cases where
$t\in \{2, 3\}$
stand out, as there are congruence classes where such counts are zero. To obtain these distributions, we obtain analytic results of independent interest. We determine the asymptotics, near roots of unity, of the ubiquitous infinite products