Fibrations meromorphes sur certaines varietes de classe canonique triviale
Abstract
Motivated by an example, due to Voisin, of a smooth simply-connected projective variety with trivial canonical class and cyclic Picard group, admitting a meromorphic endomorphism of high degree, we study meromorphic fibrations on certain varieties with trivial canonical class. We show (theorem 2.1) that any nontrivial meromorphic fibration of a variety with trivial canonical class and cyclic Picard group is in varieties of general type. Therefore, Voisin's endomorphism cannot preserve a fibration, at least in the generic case. As this example is symplectic, we study meromorphic fibrations on symplectic varieties. In dimension 4 (and in all dimensions modulo minimal model program) we get some results similar to Matsushita's. The main result here is theorem 3.6. We also study the relations between meromorphic endomorphisms and meromorphic fibrations. In particular, we prove that if an endomorphism does not preserve a fibration, then its general iterated orbit is Zariski-dense. We show that some power of an endomorphism preserves the "core" (a natural fibration constructed by the second author), and ask the similar question about the Iitaka fibration.
Keywords
Cite
@article{arxiv.math/0510299,
title = {Fibrations meromorphes sur certaines varietes de classe canonique triviale},
author = {Ekaterina Amerik and Frederic Campana},
journal= {arXiv preprint arXiv:math/0510299},
year = {2016}
}
Comments
28 pages, in French, LaTeX. v2, v3: minor grammatical changes. v4: a proposition (4.8) on potential density over function fields, implicit in the earlier versions, is stated explicitely. v5: the converse to prop. 3.1 added (second part of prop. 3.1), together with a general result implying it (theorem 2.3)