We introduce the notion of refined unramified cohomology of algebraic schemes and prove comparison theorems that identify some of these groups with cycle groups. This generalizes to cycles of arbitrary codimension previous results of Bloch–Ogus, Colliot-Thélène–Voisin, Kahn, Voisin, and Ma. We combine our approach with the Bloch–Kato conjecture, proven by Voevodsky, to show that on a smooth complex projective variety, any homologically trivial torsion cycle with trivial Abel–Jacobi invariant has coniveau $1$. This establishes a torsion version of a conjecture of Jannsen originally formulated $\otimes \mathbb {Q}$. We further show that the group of homologically trivial torsion cycles modulo algebraic equivalence has a finite filtration (by coniveau) such that the graded quotients are determined by higher Abel–Jacobi invariants that we construct. This may be seen as a variant for torsion cycles modulo algebraic equivalence of a conjecture of Green. We also prove $\ell$-adic analogues of these results over any field $k$ which contains all $\ell$-power roots of unity.

We develop the theory of relative regular holonomic $\mathcal {D}$-modules with a smooth complex manifold $S$ of arbitrary dimension as parameter space, together with their main functorial properties. In particular, we establish in this general setting the relative Riemann–Hilbert correspondence proved in a previous work in the one-dimensional case.

We define Bernstein–Sato polynomials for meromorphic functions and study their basic properties. In particular, we prove a Kashiwara–Malgrange-type theorem on their geometric monodromies, which would also be useful in relation with the monodromy conjecture. A new feature in the meromorphic setting is that we have several b-functions whose roots yield the same set of the eigenvalues of the Milnor monodromies. We also introduce multiplier ideal sheaves for meromorphic functions and show that their jumping numbers are related to our b-functions.

Designing an algorithm with a singly exponential complexity for computing semialgebraic triangulations of a given semialgebraic set has been a holy grail in algorithmic semialgebraic geometry. More precisely, given a description of a semialgebraic set $S \subset \mathbb {R}^k$ by a first-order quantifier-free formula in the language of the reals, the goal is to output a simplicial complex $\Delta $, whose geometric realization, $|\Delta |$, is semialgebraically homeomorphic to S. In this paper, we consider a weaker version of this question. We prove that for any $\ell \geq 0$, there exists an algorithm which takes as input a description of a semialgebraic subset $S \subset \mathbb {R}^k$ given by a quantifier-free first-order formula $\phi $ in the language of the reals and produces as output a simplicial complex $\Delta $, whose geometric realization, $|\Delta |$ is $\ell $-equivalent to S. The complexity of our algorithm is bounded by $(sd)^{k^{O(\ell )}}$, where s is the number of polynomials appearing in the formula $\phi $, and d a bound on their degrees. For fixed $\ell $, this bound is singly exponential in k. In particular, since $\ell $-equivalence implies that the homotopy groups up to dimension $\ell $ of $|\Delta |$ are isomorphic to those of S, we obtain a reduction (having singly exponential complexity) of the problem of computing the first $\ell $ homotopy groups of S to the combinatorial problem of computing the first $\ell $ homotopy groups of a finite simplicial complex of size bounded by $(sd)^{k^{O(\ell )}}$.

We describe the Galois action on the middle $\ell $-adic cohomology of smooth, projective fourfolds $K_A(v)$ that occur as a fiber of the Albanese morphism on moduli spaces of sheaves on an abelian surface A with Mukai vector v. We show this action is determined by the action on $H^2_{\mathrm {\acute{e}t}}(A_{\bar {k}},{\mathbb Q}_{\ell }(1))$ and on a subgroup $G_A(v) \leqslant (A\times \hat {A})[3]$, which depends on v. This generalizes the analysis carried out by Hassett and Tschinkel over ${\mathbb C}$ [21]. As a consequence, over number fields, we give a condition under which $K_2(A)$ and $K_2(\hat {A})$ are not derived equivalent.

The points of $G_A(v)$ correspond to involutions of $K_A(v)$. Over ${\mathbb C}$, they are known to be symplectic and contained in the kernel of the map $\operatorname {\mathrm {Aut}}(K_A(v))\to \mathrm {O}(H^2(K_A(v),{\mathbb Z}))$. We describe this kernel for all varieties $K_A(v)$ of dimension at least $4$.

When $K_A(v)$ is a fourfold over a field of characteristic 0, the fixed-point loci of the involutions contain K3 surfaces whose cycle classes span a large portion of the middle cohomology. We examine the fixed-point locus on fourfolds $K_A(0,l,s)$ over ${\mathbb C}$ where A is $(1,3)$-polarized, finding the K3 surface to be elliptically fibered under a Lagrangian fibration of $K_A(0,l,s)$.

In this paper, we prove that klt singularities are invariant under deformations if the generic fiber is $\mathbb {Q}$-Gorenstein. We also obtain a similar result for slc singularities. These are generalizations of results of Esnault-Viehweg [Math. Ann. 271 (1985), 439–449] and S. Ishii [Math. Ann. 275 (1986), 139–148; Singularities (Iowa City, IA, 1986) Contemporary Mathematics, vol. 90 (American Mathematical Society, Providence, RI, 1989), 135–145].

The goal of this note is to show that in the case of ‘transversal intersections’ the ‘true local terms’ appearing in the Lefschetz trace formula are equal to the ‘naive local terms’. To prove the result, we extend the strategy used in our previous work, where the case of contracting correspondences is treated. Our new ingredients are the observation of Verdier that specialization of an étale sheaf to the normal cone is monodromic and the assertion that local terms are ‘constant in families’. As an application, we get a generalization of the Deligne–Lusztig trace formula.

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. IHES 129 (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.

Abstract prepared by Rodrigo Figueiredo.

E-mail: rodrigo@ime.usp.br

URL: https://doi.org/10.11606/T.45.2019.tde-28042019-181150

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.