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We show that the natural morphism $ϕ : π_{1} (X_{η}, x_{η}) \to π_{1} {(X, x)}_{η}$ between the fundamental group scheme of the generic fiber $X_{η}$ of a scheme $X$ over a connected Dedekind scheme and the generic fiber of the fundamental group scheme of $X$ is always faithfully flat. As an application we give a necessary and sufficient condition for a finite, dominated pointed $G$ -torsor over $X_{η}$ to be extended over $X$ . We finally provide examples where $ϕ : π_{1} (X_{η}, x_{η}) \to π_{1} {(X, x)}_{η}$ is an isomorphism.

Comparison theorems for Gromov–Witten invariants of smooth pairs and of degenerations

Dan Abramovich, Steffen Marcus, Jonathan Wise (2014)

Annales de l’institut Fourier

We consider four approaches to relative Gromov–Witten theory and Gromov–Witten theory of degenerations: J. Li’s original approach, B. Kim’s logarithmic expansions, Abramovich–Fantechi’s orbifold expansions, and a logarithmic theory without expansions due to Gross–Siebert and Abramovich–Chen. We exhibit morphisms relating these moduli spaces and prove that their virtual fundamental classes are compatible by pushforward through these morphisms. This implies that the Gromov–Witten invariants associated...

Complete Kähler-Einstein Metrics on Bounded Domains Locally of Finite Volume at Some Boundary Points.

Ngaiming Mok (1988)

Mathematische Annalen

Conducteur, descente et pincement

Daniel Ferrand (2003)

Bulletin de la Société Mathématique de France

Une somme amalgamée de schémas est décrite localement par un produit fibré d’anneaux. Ce texte donne un résultat global d’existence (§5.4) de schémas définis comme certaines sommes amalgamées et un procédé algébrique (§2.2) pour décrire les modules sur produits fibrés d’anneaux correspondants.