Fundamental groups of rationally connected varieties.

*(English)*Zbl 1077.14520The notion of a rationally connected variety was introduced by the author, Y. Miyaoka and S. Mori [J. Algebr. Geom. 1, 429–448 (1992; Zbl 0780.14026)] and says that any two points of a proper variety \(X\) over a field of characteristic \(0\) are in the image of an appropriate morphism \(f:\mathbb{P}^1\to X\).

By a result of J.-P. Serre [J. Lond. Math. Soc. 34, 481–484 (1959; Zbl 0097.36301)], it turns out that rationally connected varieties are simply connected. In the paper under review, the author investigates the more sophisticated problem of the fundamental groups of open subsets in a rationally connected variety. The main result is the following

Theorem. Let \(X\) be a smooth projective rationally connected variety \(X\) over an algebraically closed field of characteristic \(0\), let \(U\subset X\) be an open subset, and let \(x_0\in U\) be a point. Then there is an open subset \(0\in V\subset {\mathbb{A}}^1\) and a morphism \(f: V\to U\) such that \(f(0)=x_0\), and \(\pi_1(V,0)\to\pi_1(U,x_0)\) is surjective.

Several applications to \(p\)-adic fields and the image of the monodromy representations for the moduli space of genus \(g\) curves (which is known to be unirational for \(2\leq g \leq 13\)) are given.

By a result of J.-P. Serre [J. Lond. Math. Soc. 34, 481–484 (1959; Zbl 0097.36301)], it turns out that rationally connected varieties are simply connected. In the paper under review, the author investigates the more sophisticated problem of the fundamental groups of open subsets in a rationally connected variety. The main result is the following

Theorem. Let \(X\) be a smooth projective rationally connected variety \(X\) over an algebraically closed field of characteristic \(0\), let \(U\subset X\) be an open subset, and let \(x_0\in U\) be a point. Then there is an open subset \(0\in V\subset {\mathbb{A}}^1\) and a morphism \(f: V\to U\) such that \(f(0)=x_0\), and \(\pi_1(V,0)\to\pi_1(U,x_0)\) is surjective.

Several applications to \(p\)-adic fields and the image of the monodromy representations for the moduli space of genus \(g\) curves (which is known to be unirational for \(2\leq g \leq 13\)) are given.

Reviewer: Olaf Teschke (Berlin)