Integrality in topology stemming from complex algebraic varieties
Người báo cáo: Hélène Esnault (Berlin, Harvard and Copenhagen)
Thời gian: Bắt đầu từ 9h30, thứ Sáu, ngày 13/6/2025
Địa điểm: Hội trường Hoàng Tụy, Tầng 2, Nhà A6, Viện Toán học
Tóm tắt báo cáo: A smooth complex quasi-projective variety $X$ is, as a topological space, homotopic to a bouquet of spheres. In particular its fundamental group $\pi_1^{\rm top}(X)$ is finitely presented, thus finitely generated and the relations themselves are finitely generated. For example, if $X$ is an affine curve, $\pi_1^{\rm top}(X)$ is a free group. Free groups possess linear integral representations into $GL_r(r, \bar{\mathbb{Z}})$ for all $r$. Not every finitely presented group possesses linear integral representations. So we could ask whether this property is an obstruction for a finitely presented group to come from geometry, that is to be isomorphic to $\pi_1^{\rm top}(X)$ for some $X$ as above. While we have at disposal obstructions which stem from Hodge theory, such as harmonic analysis, when $X$ is projective (or proper), they do not apply for $X$ quasi-projective.
We showed (with Johan de Jong, partially based on earlier work with Michael Groechenig), that a weaker integrality notion, which we called weak integrality, is an obstruction for a finitely presented group to be geometric. The proof is now purely arithmetic (with some algebraic geometry) and relies on the Langlands correspondence. It would be of interest to know whether integrality itself (without ‘weak’) is an obstruction, we do not know this.
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