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Hélène Esnault

Publications and source records attributed to Hélène Esnault.

At least 55 records · Page 3Linked to original sources
arXiv2016

Convergent isocrystals on simply connected varieties

It is conjectured by de Jong that, if $X$ is a connected smooth projective variety over an algebraically closed field $k$ of characteristic $p>0$ with trivial étale fundamental group, any isocrystal on on $X/W$ is trivial. We prove this conjecture under two additional assumptions. Version 2: the main change is an addendum. We prove that if $X$ is a connected smooth projective variety over an algebraically closed field $k$ of characteristic $p>0$ with trivial étale fundamental group, any infinitesimal isocrystal on $X/W$ is trivial. To this aim we wrote some general facts on such infinitesimal isocrystals over W which are missing in the literature.

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arXiv2015

On the cycle class map for zero-cycles over local fields

We study the Chow group of zero-cycles of smooth projective varieties over local and strictly local fields. We prove in particular the injectivity of the cycle class map to integral l-adic cohomology for a large class of surfaces with positive geometric genus, over local fields of residue characteristic different from l. The same statement holds for semistable K3 surfaces defined over C((t)), but does not hold in general for surfaces over strictly local fields.

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arXiv2015

Chern classes of crystals

The crystalline Chern classes of the value of a locally free crystal vanish on a smooth variety defined over a perfect field. Out of this we conclude new cases of de Jong's conjecture relating the geometric étale fundamental group of a smooth projective variety defined over a perfect field and the triviality of its category of isocrystals. We also discuss the case of the Gauß-Manin convergent isocrystal.

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arXiv2015

Lefschetz theorems for tamely ramified coverings

As is well known, the Lefschetz theorems for the étale fundamental group of SGA1 do not hold. We fill a small gap in the literature showing they do for tame coverings. Let $X$ be a regular projective variety over a field $k$, and let $D\hookrightarrow X$ be a strict normal crossings divisor. Then, if $Y$ is an ample regular hyperplane intersecting $D$ transversally, the restriction functor from tame étale coverings of $X\setminus D$ to those of $Y\setminus D\cap Y$ is an equivalence if dimension $X \ge 3$, and fully faithful if dimension $X=2$. The method is dictated by Grothendieck-Murre ("The tame fundamental group of a formal neighbourhood of a divisor with normal crossings on a scheme", Springer LNM 208). The authors showed that one can lift tame coverings from $Y\setminus D\cap Y$ to the complement of $D\cap Y$ in the formal completion of $X$ along $Y$. One has then to further lift to $X\setminus D$.

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arXiv2015

Simply connected varieties in characteristic $p>0$

We show that there are no non-trivial stratified bundles over a smooth simply connected quasi-projective variety over the algebraic closure of a finite field, if the variety admits a normal projective compactification with boundary locus of codimension $\ge 2$. In the appendix, various strong forms of the Lefschetz LEF are proven. Final version, to appear in Compositio.

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arXiv2015

Index of varieties over Henselian fields and Euler characteristic of coherent sheaves

Let X be a smooth proper variety over the quotient field of a Henselian discrete valuation ring with algebraically closed residue field of characteristic p. We show that for any coherent sheaf E on X, the index of X divides the Euler-Poincaré characteristic χ(X,E) if p=0 or p>dim(X)+1. If 0 dim(X)+1). When p=0, such statements also have implications for the possible multiplicities of singular fibers in degenerations of complex projective varieties.

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arXiv2015

Non-liftability of automorphism groups of a K3 surface in positive characteristic

We show that a characteristic $0$ model $X_R\to \Spec R$, with Picard number $1$ over a geometric generic point, of a K3 surface in characteristic $p\ge 3$, essentially kills all automorphisms (Theorem 5.1). We show that there is an explicitely constructed automorphism on a supersingular K3 surface in characteristic $3$, which has positive entropy, the logarithm of a Salem number of degree $22$ (Theorem 6.4). In particular it does not lift to characteristic $0$. In addition, we show that in any large characteristic, there is an automorphism of a supersingular K3 which has positive entropy and does not lift to characteristic $0$ (Theorem 7.5).

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arXiv2014

E_1-degeneration of the irregular Hodge filtration (with an appendix by Morihiko Saito)

For a regular function f on a smooth complex quasi-projective variety, J.-D. Yu introduced in arXiv:1203.2338 a filtration (the irregular Hodge filtration) on the de Rham complex with twisted differential d+df, extending a definition of Deligne in the case of curves. In this article, we show the degeneration at E_1 of the spectral sequence attached to the irregular Hodge filtration, by using the method of arXiv:0804.4328. We also make explicit the relation with a complex introduced by M. Kontsevich and give details on its proof of the corresponding E_1 degeneration, by reduction to characteristic p, when the pole divisor of the function is reduced with normal crossings. In Appendix E, M. Saito gives a different proof of the latter statement with a possibly non reduced normal crossing pole divisor.

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arXiv2014

Deformation of algebraic cycle classes in characteristic zero

We study a formal deformation problem for rational algebraic cycle classes motivated by Grothendieck's variational Hodge conjecture. We argue that there is a close connection between the existence of a Chow-Künneth decomposition and the existence of expected deformations of cycles. This observation applies in particular to abelian schemes.

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arXiv2013

p-adic deformation of algebraic cycle classes

We study the p-adic deformation properties of algebraic cycle classes modulo rational equivalence. We show that the crystalline Chern character of a vector bundle lies in a certain part of the Hodge filtration if and only if, rationally, the class of the vector bundle lifts to a formal pro-class in K-theory on the p-adic scheme.

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arXiv2012

A finiteness theorem for Galois representations of function fields over finite fields (after Deligne)

Revised: just some typos, reorganized a bit the article. It will be published in the VIASM Annual meeting, Hanoi. We give a detailed account of Deligne's letter to Drinfeld dated June 18, 2011, in which he shows that there are finitely many irreducible lisse $\bar \Q_\ell$-sheaves with bounded ramification, up to isomorphism and up to twist, on a smooth variety defined over a finite field. The proof relies on Lafforgue's Langlands correspondence over curves. In addition, Deligne shows the existence of affine moduli of finite type over $\mathbb{Q}$. A corollary of Deligne's finiteness theorem is the existence of a number field which contains all traces of the Frobenii at closed points, which was the main result of his recent article and which answers positively his own conjecture from Weil II.

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arXiv2011

Stratified bundles and étale fundamental group (new version)

This submission replaces the arXiv:1012.5381 submission with the same title, which had been withdrawn as it contained a mistake, repaired in this submission: on $X$ projective smooth over an algebraically closed field of characteristic $p>0$, we show that all irreducible stratified bundles have rank 1 if and only if the commutator $[π_1, π_1]$ of the étale fundamental group $π_1$ is a pro-$p$-group, and we show that the category of stratified bundles is semi-simple with irreducible objects of rank 1 if and only if $ π_1 $ is abelian without $p$-power quotient. This answers positively a conjecture by Gieseker.

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arXiv2011

Rational points over finite fields for regular models of algebraic varieties of Hodge type $\geq 1$

Let $R$ be a discrete valuation ring of mixed characteristics $(0,p)$, with finite residue field $k$ and fraction field $K$, let $k'$ be a finite extension of $k$, and let $X$ be a regular, proper and flat $R$-scheme, with generic fibre $X_K$ and special fibre $X_k$. Assume that $X_K$ is geometrically connected and of Hodge type $\geq 1$ in positive degrees. Then we show that the number of $k'$-rational points of $X$ satisfies the congruence $|X(k')| \equiv 1$ mod $|k'|$. Thanks to \cite{BBE07}, we deduce such congruences from a vanishing theorem for the Witt cohomology groups $H^q(X_k, W\sO_{X_k,\Q})$, for $q > 0$. In our proof of this last result, a key step is the construction of a trace morphism between the Witt cohomologies of the special fibres of two flat regular $R$-schemes $X$ and $Y$ of the same dimension, defined by a surjective projective morphism $f : Y \to X$.

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arXiv2011

Stratified bundles and étale fundamental group

v2: A few typos corrected, a few formulations improved. On $X$ projective smooth over an algebraically closed field of characteristic $p>0$, we show that irreducible stratified bundles have rank 1 if and only if the commutator $[π_1^{{\rm \acute{e}t}}, π_1^{{\rm \acute{e}t}}]$ of the étale fundamental group is a pro-$p$-group, and we show that the category of stratified bundles is semi-simple with irreducible objects of rank 1 if and only if $ π_1^{{\rm \acute{e}t}}$ is abelian without $p$-power quotient. This answers positively a conjecture by Gieseker.

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arXiv2011

Algebraic versus topological entropy for surfaces over finite fields

We show that, as in de Rham cohomology over the complex numbers, the value of the entropy of an automorphism of the surface over a finite field $\F_q $ is taken on the span of the Néron-Severi group inside of $\ell$-adic cohomology. v2: (some) typos removed, exposition (partly) improved.

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