Maps from K-trivial varieties and connectedness problems

Lazić, Vladimir; Peternell, Thomas

Annales Henri Lebesgue, Volume 3 (2020), pp. 473-500.

In this paper we study varieties covered by rational or elliptic curves. First, we show that images of Calabi–Yau or irreducible symplectic varieties under rational maps are almost always rationally connected. Second, we investigate elliptically connected and elliptically chain connected varieties, and varieties swept out by a family of elliptic curves. Among other things, we show that Calabi–Yau or hyperkähler manifolds which are covered by a family of elliptic curves contain uniruled divisors and that elliptically chain connected varieties of dimension at least $2$ contain a rational curve, and so do $K$-trivial varieties with finite fundamental group which are covered by elliptic curves.

Dans cet article, nous étudions les variétés couvertes par des courbes rationnelles ou elliptiques. Nous montrons tout d’abord que les images des variétés de Calabi–Yau ou irréductibles symplectiques par des applications rationnelles sont presque toujours rationnellement connexes. Nous étudions ensuite les variétés elliptiquement connexes, elliptiquement connexes par chaînes ainsi que les variétés balayées par une famille de courbes elliptiques. Entre autres choses, nous montrons que les variétés de Calabi–Yau ou hyperkählériennes qui sont couvertes par une famille de courbes elliptiques contiennent des diviseurs uniréglés, et que les variétés elliptiquement connexes par chaînes de dimension au moins $2$ contiennent une courbe rationnelle, tout comme les variétés à fibré canonique trivial de groupe fondamental fini qui sont couvertes par des courbes elliptiques.

Classification: 14J32, 14H10
Keywords: Calabi–Yau varieties, families of elliptic curves, elliptically (chain) connected varieties

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