# Survey on some aspects of Lefschetz theorems in algebraic geometry

@article{Esnault2016SurveyOS, title={Survey on some aspects of Lefschetz theorems in algebraic geometry}, author={H'elene Esnault}, journal={Revista Matem{\'a}tica Complutense}, year={2016}, volume={30}, pages={217-232} }

We survey classical material around Lefschetz theorems for fundamental groups, and show the relation to parts of Deligne’s program in Weil II.

## 12 Citations

### Parabolicity conjecture of $F$-isocrystals

- Mathematics
- 2020

. In this article we prove Crew’s parabolicity conjecture of F -isocrystals. For this pur-pose, we introduce and study the notion of † -hull of a sub- F -isocrystal. On the way, we prove a new…

### Finite presentation of the tame fundamental group

- MathematicsSelecta Mathematica
- 2022

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…

### Abelian varieties not isogenous to Jacobians over global fields

- Mathematics, Computer Science
- 2021

It is proved the existence of abelian varieties not isogenous to Jacobians over characterstic p function fields by studying the action of degree p Hecke operators on hypersymmetric points using Serre Tate co-ordinates.

### Brauer groups and \'etale homotopy type

- Mathematics
- 2020

Extending a result of Schröer on a Grothendieck question in the context of complex analytic spaces, we prove that the surjectivity of the Brauer map δ : Br(X) → H ét(X,Gm,X)tor for algebraic schemes…

### The monodromy groups of lisse sheaves and overconvergent F-isocrystals

- Mathematics
- 2017

It has been proven by Serre, Larsen-Pink and Chin, that over a smooth curve over a finite field, the monodromy groups of compatible semi-simple pure lisse sheaves have "the same" $\pi_0$ and neutral…

### Etale and crystalline companions, I

- Mathematics
- 2018

Let X be a smooth scheme over a finite field of characteristic p. Consider the coefficient objects of locally constant rank on X in l-adic Weil cohomology: these are lisse Weil sheaves etale…

### Independence of algebraic monodromy groups in compatible systems

- Mathematics, Computer ScienceSelecta Mathematica
- 2021

A general method to prove independence of algebraic monodromy groups in compatible systems of representations is developed, and it is applied to deduce independence results for compatible systems both in automorphic and in positive characteristic settings.

### FACULTAD DE MATEMÁTICAS ON THE GEOGRAPHY OF SURFACES OF GENERAL TYPE WITH FIXED FUNDAMENTAL GROUP POR

- Mathematics
- 2020

In this thesis, we study the geography of complex surfaces of general type with respect to the topological fundamental group. The understanding of this general problem can be coarsely divided into…

### Rank 2 local systems and abelian varieties

- Mathematics
- 2018

Let $X/\mathbb{F}_{q}$ be a smooth, geometrically connected, quasiprojective variety. Let $\mathcal{E}$ be a semisimple overconvergent $F$-isocrystal on $X$. Suppose that irreducible summands…

## References

SHOWING 1-10 OF 55 REFERENCES

### Algebraic Topology

- Mathematics

The focus of this paper is a proof of the Nielsen-Schreier Theorem, stating that every subgroup of a free group is free, using tools from algebraic topology.

### On different notions of tameness in arithmetic geometry

- Mathematics
- 2008

The notion of a tamely ramified covering is canonical only for curves. Several notions of tameness for coverings of higher dimensional schemes have been used in the literature. We show that all these…

### Galois groups and fundamental groups

- Mathematics
- 2009

Foreword 1. Galois theory of fields 2. Fundamental groups in topology 3. Riemann surfaces 4. Fundamental groups of algebraic curves 5. Fundamental groups of schemes 6. Tannakian fundamental groups…

### $p$ -ADIC HODGE THEORY FOR RIGID-ANALYTIC VARIETIES

- MathematicsForum of Mathematics, Pi
- 2013

Abstract We give proofs of de Rham comparison isomorphisms for rigid-analytic varieties, with coefficients and in families. This relies on the theory of perfectoid spaces. Another new ingredient is…

### Cut-by-curves criterion for the overconvergence of p-adic differential equations

- Mathematics
- 2009

In this paper, we prove a ‘cut-by-curves criterion’ for the overconvergence of integrable connections on certain smooth rigid analytic spaces and certain smooth varieties over p-adic fields.

### A Lefschetz theorem for overconvergent isocrystals with Frobenius structure

- Mathematics
- 2016

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

### Langlands correspondence for isocrystals and the existence of crystalline companions for curves

- Mathematics
- 2013

In this paper, we show the Langlands correspondence for isocrystals on curves. This shows the existence of crystalline companion in the curve case. For the proof, we construct the theory of…

### Lectures on Riemann Surfaces

- Mathematics
- 1999

Contents: Covering Spaces.- Compact Riemann Surfaces.- Non-compact Riemann Surfaces.- Appendix.- References.- Symbol Index.- Author and Subject Index.

### A FINITENESS THEOREM FOR GALOIS REPRESENTATIONS OF FUNCTION FIELDS OVER FINITE FIELDS (AFTER DELIGNE)

- Mathematics
- 2012

We give a detailed account of Deligne's letter (13) to Drinfeld dated June 18, 2011, in which he shows that there are nitely many irreducible lisse Q'-sheaves with bounded ramica- tion, up to…

### Fields and Galois Theory

- Mathematics
- 2006

Rings and Fields.- Integral Domains and Polynomials.- Field Extensions.- Applications to Geometry.- Splitting Fields.- Finite Fields.- The Galois Group.- Equations and Groups.- Some Group Theory.-…