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Building upon work of Epstein, May and Drury, we define and investigate the mod p Steenrod operations on the de Rham cohomology of smooth algebraic stacks over a field of characteristic $p>0$. We then compute the action of the operations on the de Rham cohomology of classifying stacks for finite groups, connected reductive groups for which p is not a torsion prime and (special) orthogonal groups when $p=2$.
We give an arithmetic count of the lines on a smooth cubic surface over an arbitrary field $k$, generalizing the counts that over ${\mathbf {C}}$ there are $27$ lines, and over ${\mathbf {R}}$ the number of hyperbolic lines minus the number of elliptic lines is $3$. In general, the lines are defined over a field extension $L$ and have an associated arithmetic type $\alpha$ in $L^*/(L^*)^2$. There is an equality in the Grothendieck–Witt group $\operatorname {GW}(k)$ of $k$,
where $\operatorname {Tr}_{L/k}$ denotes the trace $\operatorname {GW}(L) \to \operatorname {GW}(k)$. Taking the rank and signature recovers the results over ${\mathbf {C}}$ and ${\mathbf {R}}$. To do this, we develop an elementary theory of the Euler number in $\mathbf {A}^1$-homotopy theory for algebraic vector bundles. We expect that further arithmetic counts generalizing enumerative results in complex and real algebraic geometry can be obtained with similar methods.
We define a motivic conductor for any presheaf with transfers F using the categorical framework developed for the theory of motives with modulus by Kahn, Miyazaki, Saito and Yamazaki. If F is a reciprocity sheaf, this conductor yields an increasing and exhaustive filtration on
$F(L)$, where L is any henselian discrete valuation field of geometric type over the perfect ground field. We show that if F is a smooth group scheme, then the motivic conductor extends the Rosenlicht–Serre conductor; if F assigns to X the group of finite characters on the abelianised étale fundamental group of X, then the motivic conductor agrees with the Artin conductor defined by Kato and Matsuda; and if F assigns to X the group of integrable rank
$1$ connections (in characteristic
$0$), then it agrees with the irregularity. We also show that this machinery gives rise to a conductor for torsors under finite flat group schemes over the base field, which we believe to be new. We introduce a general notion of conductors on presheaves with transfers and show that on a reciprocity sheaf, the motivic conductor is minimal and any conductor which is defined only for henselian discrete valuation fields of geometric type with perfect residue field can be uniquely extended to all such fields without any restriction on the residue field. For example, the Kato–Matsuda Artin conductor is characterised as the canonical extension of the classical Artin conductor defined in the case of a perfect residue field.
We prove that the special-value conjecture for the zeta function of a proper, regular, flat arithmetic surface formulated in [6] at $s=1$ is equivalent to the Birch and Swinnerton-Dyer conjecture for the Jacobian of the generic fibre. There are two key results in the proof. The first is the triviality of the correction factor of [6, Conjecture 5.12], which we show for arbitrary regular proper arithmetic schemes. In the proof we need to develop some results for the eh-topology on schemes over finite fields which might be of independent interest. The second result is a different proof of a formula due to Geisser, relating the cardinalities of the Brauer and the Tate–Shafarevich group, which applies to arbitrary rather than only totally imaginary base fields.
We introduce irregular constructible sheaves, which are ${\mathbb {C}}$-constructible with coefficients in a finite version of the Novikov ring $\Lambda$ and special gradings. We show that the bounded derived category of cohomologically irregular constructible complexes is equivalent to the bounded derived category of holonomic ${\mathcal {D}}$-modules by a modification of D’Agnolo and Kashiwara's irregular Riemann–Hilbert correspondence. The bounded derived category of cohomologically irregular constructible complexes is equipped with the irregular perverse $t$-structure, which is a straightforward generalization of usual perverse $t$-structure, and we prove that its heart is equivalent to the abelian category of holonomic ${\mathcal {D}}$-modules. We also develop the algebraic version of the theory.
Following the work of Mustaţă and Bitoun, we recently developed a notion of Bernstein–Sato roots for arbitrary ideals, which is a prime characteristic analogue for the roots of the Bernstein–Sato polynomial. Here, we prove that for monomial ideals the roots of the Bernstein–Sato polynomial (over $\mathbb{C}$) agree with the Bernstein–Sato roots of the mod $p$ reductions of the ideal for $p$ large enough. We regard this as evidence that the characteristic-$p$ notion of Bernstein–Sato root is reasonable.
Let $\mathcal {F}$ be a polystable sheaf on a smooth minimal projective surface of Kodaira dimension 0. Then the differential graded (DG) Lie algebra $R\operatorname {Hom}(\mathcal {F},\mathcal {F})$ of derived endomorphisms of $\mathcal {F}$ is formal. The proof is based on the study of equivariant $L_{\infty }$ minimal models of DG Lie algebras equipped with a cyclic structure of degree 2 which is non-degenerate in cohomology, and does not rely (even for K3 surfaces) on previous results on the same subject.
In this paper we prove a categorification of the Grothendieck–Riemann–Roch theorem. Our result implies in particular a Grothendieck–Riemann–Roch theorem for Toën and Vezzosi's secondary Chern character. As a main application, we establish a comparison between the Toën–Vezzosi Chern character and the classical Chern character, and show that the categorified Chern character recovers the classical de Rham realization.
We prove the analog of the Morel–Voevodsky localization theorem for framed motivic spaces. We deduce that framed motivic spectra are equivalent to motivic spectra over arbitrary schemes, and we give a new construction of the motivic cohomology of arbitrary schemes.
We study log $\mathscr {D}$-modules on smooth log pairs and construct a comparison theorem of log de Rham complexes. The proof uses Sabbah’s generalized b-functions. As applications, we deduce a log index theorem and a Riemann-Roch type formula for perverse sheaves on smooth quasi-projective varieties. The log index theorem naturally generalizes the Dubson-Kashiwara index theorem on smooth projective varieties.
In this article we introduce the local versions of the Voevodsky category of motives with $\mathbb{F} _p$-coefficients over a field k, parametrized by finitely generated extensions of k. We introduce the so-called flexible fields, passage to which is conservative on motives. We demonstrate that, over flexible fields, the constructed local motivic categories are much simpler than the global one and more reminiscent of a topological counterpart. This provides handy ‘local’ invariants from which one can read motivic information. We compute the local motivic cohomology of a point for $p=2$ and study the local Chow motivic category. We introduce local Chow groups and conjecture that over flexible fields these should coincide with Chow groups modulo numerical equivalence with$\mathbb{F} _p$-coefficients, which implies that local Chow motives coincide with numerical Chow motives. We prove this conjecture in various cases.
We study relationships between the Nisnevich topology on smooth schemes and certain Grothendieck topologies on proper and not necessarily proper modulus pairs, which were introduced in previous papers. Our results play an important role in the theory of sheaves with transfers on proper modulus pairs.
We show that the only summands of the theta divisor on Jacobians of curves and on intermediate Jacobians of cubic threefolds are the powers of the curve and the Fano surface of lines on the threefold. The proof only uses the decomposition theorem for perverse sheaves, some representation theory and the notion of characteristic cycles.
Using Auroux’s description of Fukaya categories of symmetric products of punctured surfaces, we compute the partially wrapped Fukaya category of the complement of
$k+1$
generic hyperplanes in
$\mathbb{CP}^{n}$
, for
$k\geqslant n$
, with respect to certain stops in terms of the endomorphism algebra of a generating set of objects. The stops are chosen so that the resulting algebra is formal. In the case of the complement of
$n+2$
generic hyperplanes in
$\mathbb{C}P^{n}$
(
$n$
-dimensional pair of pants), we show that our partial wrapped Fukaya category is equivalent to a certain categorical resolution of the derived category of the singular affine variety
$x_{1}x_{2}\ldots x_{n+1}=0$
. By localizing, we deduce that the (fully) wrapped Fukaya category of the
$n$
-dimensional pair of pants is equivalent to the derived category of
$x_{1}x_{2}\ldots x_{n+1}=0$
. We also prove similar equivalences for finite abelian covers of the
$n$
-dimensional pair of pants.
Let K be a complete discrete valuation field of characteristic
$0$
, with not necessarily perfect residue field of characteristic
$p>0$
. We define a Faltings extension of
$\mathcal {O}_K$
over
$\mathbb {Z}_p$
, and we construct a Hodge-Tate filtration for abelian varieties over K by generalizing Fontaine’s construction [Fon82] where he treated the perfect residue field case.
We correct the proof of the main
$\ell$
-independence result of the above-mentioned paper by showing that for any smooth and proper variety over an equicharacteristic local field, there exists a globally defined such variety with the same (
$p$
-adic and
$\ell$
-adic) cohomology.
We give a list of statements on the geometry of elliptic threefolds phrased only in the language of topology and homological algebra. Using only notions from topology and homological algebra, we recover existing results and prove new results on torsion pairs in the category of coherent sheaves on an elliptic threefold.
Given a perfect valuation ring
$R$
of characteristic
$p$
that is complete with respect to a rank-
$1$
nondiscrete valuation, we show that the ring
$\mathbb{A}_{\inf }$
of Witt vectors of
$R$
has infinite Krull dimension.
We give a geometric interpretation of sheaf cohomology for higher degrees
$n\geq 1$
in terms of torsors on the member of degree
$d=n-1$
in hypercoverings of type
$r=n-2$
, endowed with an additional datum, the so-called rigidification. This generalizes the fact that cohomology in degree one is the group of isomorphism classes of torsors, where the rigidification becomes vacuous, and that cohomology in degree two can be expressed in terms of bundle gerbes, where the rigidification becomes an associativity constraint.
We construct a scheme
$B(r; {\mathbb {A}}^n)$
such that a map
$X \to B(r; {\mathbb {A}}^n)$
corresponds to a degree-n étale algebra on X equipped with r generating global sections. We then show that when
$n=2$
, i.e., in the quadratic étale case, the singular cohomology of
$B(r; {\mathbb {A}}^n)({\mathbb {R}})$
can be used to reconstruct a famous example of S. Chase and to extend its application to showing that there is a smooth affine
$r-1$
-dimensional
${\mathbb {R}}$
-variety on which there are étale algebras
${\mathcal {A}}_n$
of arbitrary degrees n that cannot be generated by fewer than r elements. This shows that in the étale algebra case, a bound established by U. First and Z. Reichstein in [6] is sharp.