Maps from K-trivial varieties and connectedness problems

TL;DR

This paper explores the connectedness properties of K-trivial varieties, showing that certain Calabi-Yau and hyperk"ahler manifolds covered by elliptic or rational curves exhibit specific rational connectedness features.

Contribution

It demonstrates that images of Calabi-Yau or symplectic varieties under rational maps are mostly rationally connected and investigates the structure of elliptically connected K-trivial varieties.

Findings

Images of Calabi-Yau varieties under rational maps are almost always rationally connected.

Calabi-Yau or hyperk"ahler manifolds covered by elliptic curves contain uniruled divisors.

Elliptically chain connected varieties of dimension ≥ 2 contain a rational curve.

Abstract

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 hyperk\"ahler manifolds which are covered by a family of elliptic curves contain uniruled divisors and that elliptically chain connected varieties of dimension at least two contain a rational curve, and so do K-trivial varieties with finite fundamental group which are covered by elliptic curves.