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Some implications between Grothendieck’s anabelian conjectures. (English) Zbl 1468.14046

Summary: Grothendieck gave two forms of his “main conjecture of anabelian geometry”, namely the section conjecture and the hom conjecture. He stated that these two forms are equivalent and that if they hold for hyperbolic curves, then they hold for elementary anabelian varieties too. We state a stronger form of Grothendieck’s conjecture (equivalent in the case of curves) and prove that Grothendieck’s statements hold for our form of the conjecture. We work with DM stacks, rather than schemes. If \(X\) is a DM stack over \(k\subseteq\mathbb{C}\), we prove that whether \(X\) satisfies the conjecture or not depends only on \(X_{\mathbb{C}}\). We prove that the section conjecture for hyperbolic orbicurves stated by N. Borne and M. Emsalem [Bull. Soc. Math. Fr. 142, No. 3, 465–487 (2014; Zbl 1327.14103)] follows from the conjecture for hyperbolic curves.

MSC:

14F35 Homotopy theory and fundamental groups in algebraic geometry
14F30 \(p\)-adic cohomology, crystalline cohomology
11G35 Varieties over global fields
14A20 Generalizations (algebraic spaces, stacks)

Citations:

Zbl 1327.14103
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