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We formulate and prove a non-abelian analog of Deligne's Fixed Part theorem on Hodge classes, revisiting previous work of Jost--Zuo, Katzarkov--Pantev and Landesman--Litt. To this aim we study algebraically isomonodromic extensions of local systems and we relate them to variations of Hodge structures, for example we show that the Mumford-Tate group at a generic point stays constant in an algebraically isomonodromic extension of a variation of Hodge structure. v2: a few typos ironed and Thm 1.1 5) completed. v3: there was a Schlamassel leading to a mix-up of files. Apologies. Else identical version (one minor change)., Comment: 23 pages

We study the geometry and cohomology of Lefschetz pencils for semi-stable schemes over a discrete valuation ring. We relate the global cohomological properties of the Lefschetz pencil and the monodromy-weight conjecture, in particular we show that if one assumes the monodromy-weight conjecture in smaller dimensions then one can obtain a rather complete understanding of the relative cohomology of the pencil. This reduces the monodromy-weight conjecture to an arithmetic variant of a conjecture of Kashiwara for the projective line., Comment: 47 pages

We generalise a theorem on the existence of Frobenius isocrystal and Fontaine-Laffaille module structures on rigid flat connections to the non-proper setting. The proof is based on a new strategy of a point-set topological flavour, which allows us to produce a purely $p$-adic statement and thereby to avoid the classical Simpson correspondence., Comment: corrections to Prop. 3.9, 28 pages

This is a small note on Manin's 1966 article on rational surfaces over perfect fields, the conjecture he formulates there, and later developments. This text is by no means exhaustive and reflects the author's understanding and interest. References for background and general studies are provided., Comment: 9 pages

This small text was written for the AMS Notices. It is a survey of integrality properties of complex local systems, where I tried to single out one example which is not entirely explicit in the literature. The focus is on the obstruction it yields for a finitely presented group to be the topological fundamental group of a connected smooth quasi-projective complex variety. I thank Johan de Jong, Michael Groechenig, and Moritz Kerz. The material in this expository note relies on joint work or discussions with them. This version: I have tried several times to make the AMS bib file compatible with the arXiv requirement, it never worked..., today it seems to do!, Comment: 10 pages

Those notes rest on the Samuel Eilenberg Lectures I gave at Columbia University, NY, in the fall 2022. I thank all the mathematicians who participated in their elaboration, directly or indirectly. They are meant to be published as a Springer Lecture Notes. This arXiv version is not identical with the final version., Comment: 69 pages

If in a given rank $r$, there is an irreducible complex local system with torsion determinant and quasi-unipotent monodromies at infinity on a smooth quasi-projective variety, then for every prime number $\ell$, there is an absolutely irreducible $\ell$-adic local system of the same rank, with the same determinant and monodromies at infinity, up to semi-simplification. A finitely presented group is said to be weakly integral with respect to a torsion character and a rank $r$ if once there is an irreducible rank $r$ complex linear representation, then for any $\ell$, there is an absolutely irreducible one of rank $r$ and determinant this given character which is defined over $\bar{ \mathbb{Z}}_\ell$. We prove that this property is a new obstruction for a finitely presented group to be the fundamental group of a smooth qusi-projective complex variety. The proofs rely on the arithmetic Langlands program via the existence of Deligne's companions (L. Lafforgue, Drinfeld) and the geometric Langlands program via de Jong's conjecture (Gaitsgory for $\ell \ge 3$). We also define weakly arithmetic complex local systems and show they are Zariski dense in the Betti moduli. Finally we show that our method gives an arithmetic proof of Corlette-T. Mochizuki theorem, proved using tame pure imaginary harmonic metrics, after which the pull-back by a morphism between two smooth complex algebraic varieties of a semi-simple complex local system is semi-simple. v2: a mistake pointed out by the kind referee in the proof of Theorem 7.3 is corrected. Final version: appears in Transactions AMS, Comment: 18 pages

For any even natural number $r \ge 2$, we construct an irreducible rigid non-cohomologically rigid complex local system of rank $r$ on a smooth projective variety depending on $r$. For $r=2$, we construct an irreducible rigid non-cohomogically rigid local system of rank $2$ on a quasi-projective variety which becomes cohomologically rigid after fixing the conjugacy classes of the monodromies at infinity. v2: We added a remark due to Alexander Petrov: by taking the exterior product of our examples with a (cohomologically) rigid local system with infinite monodromy, we obtain examples of rigid non-cohomologically rigid local systems with infinite monodromy., Comment: 7 pages

v2: For a projective variety defined over a finite field with $q$ elements, it is shown that as algebraic integers, the eigenvalues of the geometric Frobenius acting on $\ell$-adic cohomology have higher than known $q$-divisibility beyond the middle dimension. This sharpens both Deligne's integrality theorem and the cohomological divisibility theorem proven by the first author and N. Katz. Similar lower bounds are proved for the Hodge level for a complex variety beyond the middle dimension, improving earlier results in this direction. We discuss the affine case. The previous version contained a gap at this place. We are thankful to Dingxin Zhang for noticing it., Comment: 7 pages

The main purpose of this work is to prove the Andr\'e-Oort conjecture in full generality., Comment: Many details have been clarified and elaborated on, as well as minor errors corrected

We introduce a new obstruction to lifting smooth proper varieties in characteristic $p>0$ to characteristic $0$. It is based on Grothendieck's specialization homomorphism and the resulting discrete finiteness properties of \'etale fundamental groups., Comment: 18 pages

Let $p$ be a prime number, and let $k$ be an algebraically closed field of characteristic $p$. We show that the tame fundamental group of a smooth affine curve over $k$ is a projective profinite group. We prove that the fundamental group of a smooth projective variety over $k$ is finitely presented. More generally we prove that the tame fundamental group of a smooth quasi-projective variety over $k$ which admits a good compactification is finitely presented. v2: references added. Thank you to all for the friendly and fruitful comments., Comment: 19 pages

We show that complex local systems with quasi-unipotent monodromy at infinity over a normal complex variety are Zariski dense in their moduli. v2: we waited for feedback and added a consequence of Alexandr Petrov's theorem. 3: we tightened the last section. Final version: appears in Israel Journal of Mathematics. footnote added to Conjecture 1.1: Aaron Landesman and Daniel Litt just made available a preprint showing that there is a lower bound for the rank of geometric local systems with infinite mon-odromy on certain curves, and consequently the conjecture can not be true in this generality., Comment: 11 pages

We use the notion of universal extension in a linear abelian category to study extensions of variations of mixed Hodge structure and convergent and overconvergent isocrystals. The results we obtain apply, for example, to prove the exactness of some homotopy sequences for these categories and to study $F$-able isocrystals., Comment: 19 pages; added reference to [And20]

We prove that $\bar {\mathbb Q}_\ell$-local systems of bounded rank and ramification on a smooth variety $X$ defined over an algebraically closed field $k$ of characteristic $p\neq \ell$ are tamified outside of codimension $2$ by a finite separable cover of bounded degree. In rank one, there is a curve which preserves their monodromy. There is a curve defined over the algebraic closure of a purely transcendental extension of $k$ of finite degree which fulfills the Lefschetz theorem. Last version: minor typos corrected., Comment: 13 pages

We propose a conjecture on the density of arithmetic points in the deformation space of representations of the \'etale fundamental group in positive characteristic. This? conjecture has applications to \'etale cohomology theory, for example it implies a Hard Lefschetz conjecture. We prove the density conjecture in tame degree two for the curve $\mathbb{P}^1\setminus \{0,1,\infty\}$. v2: very small typos corrected.v3: final. Publication in Epiga., Comment: 18 pages

It is a short report for the ICCM2019 Proceedings on recent results obtained with Michael Groechenig and Moritz Kerz concerning special subloci of the Betti moduli space of irreducible complex local systems on complex varieties., Comment: 8 pages

We show that in positive characteristic special loci of deformation spaces of rank one $\ell$-adic local systems are quasilinear. From this we deduce the Hard Lefschetz theorem for rank one $\ell$-adic local systems and a generic vanishing theorem. Last version: a few typos corrected, Comment: 26 pages

We show that closed subsets of the character variety of a complex variety with negatively weighted homology, which are $p$-adically integral and Galois invariant, are motivic. Final version: Cambridge Journal of Mathematics, Comment: latex 20 pages

We construct a logarithmic model of connections on smooth quasi-projective $n$-dimensional geometrically irreducible varieties defined over an algebraically closed field of characteristic $0$. It consists of a good compactification of the variety together with $(n+1)$ lattices on it which are stabilized by log differential operators, and compute algebraically de Rham cohomology. The construction is derived from the existence of good Deligne-Malgrange lattices, a theorem of Kedlaya and Mochizuki which consists first in eliminating the turning points. Moreover, we show that a logarithmic model obtained in this way, called a good model, yields a formula predicted by Michael Groechenig, computing the class of the characteristic variety of the underlying D-module in the $K$-theory group of the variety., Comment: 24 pages. V2 25 pages, revised version. V3: final version to appear in Documenta Mathematica

We show that bounding ramification at infinity bounds fierce ramification. This answers positively a question of Deligne posed to the first named author., Comment: latex, 9 pages

We give a proof without use of perfectoid geometry of Scholzes' vanishing theorem of \'etale cohomology with $\mathbb{F}_p$-coefficients beyond the dimension of projective varieties in a specific pro $p$-tower in characteristic not equal to $p$. Final version., Comment: 7 pages

We prove that the $\ell$-adic Chern classes of canonical extensions of automorphic vector bundles, over toroidal compactifications of Shimura varieties of Hodge type over $\bar{ \mathbb{Q}}_p$, descend to classes in the $\ell$-adic cohomology of the minimal compactifications. These are invariant under the Galois group of the $p$-adic field above which the variety and the bundle are defined., Comment: 28 pages

We prove that the monodromy of an irreducible cohomologically complex rigid local system with finite determinant and quasi-unipotent local monodromies at infinity on a smooth quasiprojective complex variety $X$ is integral. This answers positively a special case of a conjecture by Carlos Simpson. On a smooth projective variety, the argument relies on Drinfeld's theorem on the existence of $\ell$-adic companions over a finite field. When the variety is quasiprojective, one has in addition to control the weights and the monodromy at infinity., Comment: 13 pages, new version includes the more general case of quasi-unipotent monodromies at infinity

An irreducible integrable connection $(E,\nabla)$ on a smooth projective complex variety $X$ is called rigid if it gives rise to an isolated point of the corresponding moduli space $\mathcal{M}_{dR}(X)$. According to Simpson's motivicity conjecture, irreducible rigid flat connections are of geometric origin, that is, arise as subquotients of a Gau\ss-Manin connection of a family of smooth projective varieties defined on an open dense subvariety of $X$. In this article we study mod $p$ reductions of irreducible rigid connections and establish results which confirm Simpson's prediction. In particular, for large $p$, we prove that $p$-curvatures of mod $p$ reductions of irreducible rigid flat connections are nilpotent, and building on this result, we construct an $F$-isocrystalline realization for {irreducible} rigid flat connections. More precisely, we prove that there exist smooth models $X_R$ and $(E_R,\nabla_R)$ of $X$ and $(E,\nabla)$, over a finite type ring $R$, such that for every Witt ring $W(k)$ of a finite field $k$ and every homomorphism $R \to W(k)$, the $p$-adic completion of the base change $(\widehat{E}_{W(k)},\widehat{\nabla}_{W(k)})$ on $\widehat{X}_{W(k)}$ represents an $F$-isocrystal. Subsequently we show that {irreducible} rigid flat connections with vanishing $p$-curvatures are unitary. This allows us to prove new cases of the Grothendieck--Katz $p$-curvature conjecture. We also prove the existence of a complete companion correspondence for $F$-isocrystals stemming from irreducible cohomologically rigid connections., Comment: final version, accepted in Acta Mathematica

We prove that the vanishing of the functoriality morphism for the \'etale fundamental group between smooth projective varieties over an algebraically closed field of characteristic $p>0$ forces the same property for the fundamental groups of stratifications. 2nd version: Acknowledgement added., Comment: 11 pages

We prove that Chern classes in continuous $\ell$-adic cohomology of automorphic bundles associated to representations of $G$ on a projective Shimura variety with data $(G,X)$ are trivial rationally. It is a consequence of Beilinson's conjectures which predict that the Chern classes in the Chow groups vanish rationally., Comment: 18 pages

We investigate an analogue of the Grothendieck $p$-curvature conjecture, where the vanishing of the $p$-curvature is replaced by the stronger condition, that the module with connection mod $p$ underlies a $\mathcal{D}_X$-module structure. We show that this weaker conjecture holds in various situations, for example if the underlying vector bundle is finite in the sense of Nori, or if the connection underlies a $\mathbb{Z}$-variation of Hodge structure. We also show isotriviality assuming a coprimality condition on certain mod $p$ Tannakian fundmental groups, which in particular resolves in the projective case a conjecture of Matzat-van der Put. v2: the well known 4.2 has been added to make the note self-contained., Comment: 9 pages

We show a Lefschetz theorem for irreducible overconvergent $F$-isocrystals on smooth varieties defined over a finite field. We derive several consequences from it.

We survey classical material around Lefschetz theorems for fundamental groups, and show the relation to parts of Deligne's program in Weil II. The notes are based on some aspects of the material for the Santal\'o lectures at the Universidad Complutense de Madrid (October 2015) and the Rademacher lectures at the University of Pennsylvania (February 2016). They appear in Revista Matem\'atica Complutense., Comment: 13 pages

Over a connected geometrically unibranch scheme $X$ of finite type over a finite field, we show finiteness of the number of irreducible $\bar \Q_\ell$-lisse sheaves, with bounded rank and bounded ramification in the sense of Drinfeld, up to twist by a character of the finite field. On $X$ smooth, with bounded ramification in the sense of bounding the Swan conductors on curves, this is Deligne's theorem. Version 2: We prove also that for Deligne's finiteness theorem, it is enough to assume $X$ normal. However, the proof uses the smooth case, unlike the proof in the case of bounded ramification in the sense of Drinfeld. Finally we discuss the various notions of boundedness of ramification used., Comment: latex, 6 pages; to appear in IMRN

Those are the notes for the 2015 Summer Research Institute on Algebraic Geometry. We report on Deligne's finiteness theorem for $\ell$-adic representations on smooth varieties defined over a finite field, on its crystalline version, and on how the geometric \'etale fundamental group of a smooth projective variety defined over a characteristic $p>0$ field controls crystals on the infinitesimal site and should control those on the crystalline site. v2: last results added to the report, and some typos corrected., Comment: 9 pages