The fundamental group of an algebraic link

We compute the fundamental group of an algebraic link. To cite this article: O. Neto, P.C. Silva, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

On base change of the fundamental group scheme

We provide for all prime numbers $p$ examples of smooth projective curves over a field of characteristic $p$ for which base change of the fundamental group scheme fails. This is intimately related to how $F$ -trivial vector bundles, i.e. bundles trivialized by a power of the Frobenius morphism, behave in (trivial) families. We conclude with a study of the behavior of $F$ -triviality in (not necessarily trivial) families.     

The fundamental group scheme of a non-reduced scheme

We extend the definition of fundamental group scheme to non-reduced schemes over any connected Dedekind scheme. Then we compare the fundamental group scheme of an affine scheme with that of its reduced part.     

The fundamental group of the complement of an algebraic curve

A criterion of Abhyankar for abelianness of the fundamental group of the complement of an algebraic curve in the complex projective plane is proved in a refined form by different methods.     

Contractions of the actions of reductive algebraic groups in arbitrary characteristic

Oblatum 25-X-1990 & 4-V-1991     

Harish-Chandra pairs for algebraic affine supergroup schemes over an arbitrary field

Over an arbitrary field of characteristic ≠ 2, we define the notion of Harish-Chandra pairs, and prove that the category of those pairs is anti-equivalent to the category of algebraic affine supergroup schemes. The result is applied to characterize some classes of affine supergroup schemes such as those which are (a) simply connected, (b) unipotent or (c) linearly reductive in positive characteristic.     

Exotic Stein fillings with arbitrary fundamental group

Let G be a finitely presentable group. We provide an infinite family of homeomorphic but pairwise non-diffeomorphic, symplectic but non-complex closed 4-manifolds with fundamental group G such that each member of the family admits a Lefschetz fibration of the same genus over the two-sphere. As a corollary, we also show the existence of a contact 3-manifold which admits infinitely many homeomorphic but pairwise non-diffeomorphic Stein fillings such that the fundamental group of each filling is isomorphic to G. Moreover, we observe that the contact 3-manifold above is contactomorphic to the link of some isolated complex surface singularity equipped with its canonical contact structure.     

Smooth structures on algebraic surfaces with cyclic fundamental group

Partially supported by NSERC grant A4000     

Smooth structures on algebraic surfaces with finite fundamental group

Oblatum 2-VIII-1988 &; 27-XII-1989     

The fundamental group of period domains over finite fields

We determine the fundamental group of period domains over finite fields. This answers a question of M. Rapoport raised in [M. Rapoport, Period domains over finite and local fields, in: Proc. Sympos. Pure Math., vol. 62, part 1, 1997, pp. 361-381].     

Two-dimensional representations of finitely generated free group over an arbitrary field

Working over an arbitrary field, we give equivalent conditions for a representation of a finitely generated free group with given traces and determinants to exist and to be reducible, respectively; also, we classify all two-dimensional representations of a finitely generated free group.     

The p-length of an arbitrary finite group

Estimates are given for the p-length of an arbitrary finite group.Translated from Matematicheskie Zametki, Vol. 14, No. 3, pp. 419–427, September, 1973.