The theorem of Maehara-Severi for maps of general type

TL;DR

This work generalizes finiteness phenomena for maps of general type to the setting of Campana’s orbifold base, proving that, for a fixed smooth source X over a field of characteristic zero, there are only finitely many equivalence classes of dominant rational maps f: X → Y with the orbifold base Δ_f of general type. The authors develop a robust framework built around neat maps, the orbifold base, and pullbacks of log-differentials, and they establish a model-independent criterion for maps to be of general type via κ(f) and the growth of T_r. A core technical advance is showing boundedness of the graphs of such maps, together with weak positivity and rigidity results, which yield finiteness both for maps to general-type targets and for log-general-type targets in the appropriate sense. These results have several notable consequences, including finiteness statements for maps to curves, abelian varieties, and K3 surfaces, and they illuminate Campana’s Bogomolov-sheaf finiteness and C-pair theory, providing a higher-dimensional extension of De Franchis/Maehara-type finiteness to orbifold settings.

Abstract

We prove a finiteness result for dominant rational maps whose orbifold base is of general type. Our finiteness result generalizes Maehara's theorem that a given variety dominates only finitely many projective varieties of general type up to birational equivalence, and also answers a question of Campana on the finiteness of Bogomolov sheaves. We give several further applications, including finiteness results for maps to curves, abelian varieties, and K3 surfaces.

Theorems & Definitions (62)

• Theorem : Maehara
• Definition 1.1
• Theorem A
• Corollary B: Campana's question
• Corollary C
• Definition 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof