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**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.

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.

Reviewer: Lars Kindler (Berlin)

### MSC:

14E22 | Ramification problems in algebraic geometry |

14H30 | Coverings of curves, fundamental group |

14F35 | Homotopy theory and fundamental groups in algebraic geometry |