## Lefschetz theorem for abelian fundamental group with modulus.(English)Zbl 1318.14014

Let $$U$$ be a smooth projective variety over a perfect field. When studying an invariant of $$U$$ it is a common technique to compare it to the same invariant of a hypersurface section $$Y$$ of $$U$$ and thus pass to studying the invariant on the lower-dimensional variety $$Y$$. One instance of such a comparison theorem is the ‘Lefschetz theorem for the étale fundamental group of $$U$$’. Proved by Grothendieck in [“Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux”, Sémin. géométrie algébrique 2 (SGA2) (1962; Zbl 0159.50402)], it roughly states that if $$\dim U\geq 2$$ (resp. $$\dim U \geq 3$$), then for a suitable hypersurface section of $$Y$$ of $$U$$, the induced map $$\pi^{\mathrm{\'et}}_1(Y)\rightarrow \pi^{\mathrm{\'et}}_1(U)$$ (omitting base points for simplicity) is surjective (resp. bijective).
If the variety $$U$$ is quasi-projective, but not necessarily projective, the analogous question is much more difficult. The article under review provides an answer, not for the full étale fundamental group, but for a certain abelian quotient.
Let $$X$$ be a smooth variety with a fixed projective embedding and let $$U$$ be an open subvariety of $$X$$, such that $$X\setminus U$$ is the support of an effective Cartier divisor $$D$$. The authors define the fundamental group $$\pi_1^{\mathrm{ab}}(X,D)$$ as the quotient of $$\pi^{\mathrm{\'et}}_1(U)$$ classifying abelian étale coverings of $$U$$ with ramification bounded by $$D$$.
The main result is the following: Assume in addition that $$X\setminus U$$ is a simple normal crossings divisor and that $$Y\subset X$$ is a hyperplane section such that $$Y\times_X (X\setminus U)$$ also is a simple normal crossings divisor. Writing $$E:=Y\times_X D$$, there is a natural map $\pi_1^{\mathrm{ab}}(Y,E)\rightarrow \pi_1^{\mathrm{ab}}(X,D).$ If $$Y$$ is sufficiently ample, then this map is surjective if $$\dim X\geq 2$$ and bijective if $$\dim X \geq 3$$. The ampleness condition on $$Y$$ depends on $$D$$, but is satisfied if the degree of $$Y$$ is sufficiently large. The proof of the theorem uses higher dimensional ramification theory developed by Brylinski, Kato and Matsuda.

### MSC:

 14E22 Ramification problems in algebraic geometry 14H30 Coverings of curves, fundamental group 14F35 Homotopy theory and fundamental groups in algebraic geometry

### Keywords:

fundamental group; ramification; Lefschetz theorem

Zbl 0159.50402
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