We show that the image of a dominant meromorphic map from an irreducible compact Calabi-Yau manifold X whose general fiber is of dimension strictly between 0 and dim X is rationally connected. Using this result, we construct for any hyper-Kähler manifold X admitting a Lagrangian fibration a Lagrangian constant cycle subvariety Σ H in X which depends on a divisor class H whose restriction to some smooth Lagrangian fiber is ample. If dim X = 4, we also show that up to a scalar multiple, the class of a zero-cycle supported on Σ H in CH 0 (X) does not depend neither on H nor on the Lagrangian fibration (provided b 2 (X) ≥ 8).

For a generalized Kummer variety X of dimension 2n, we will construct for each 0 ≤ i ≤ n some coisotropic subvarieties in X foliated by i-dimensional constant cycle subvarieties. These subvarieties serve to prove that the rational orbit filtration introduced by Voisin on the Chow group of zero-cycles of a generalized Kummer variety coincides with the induced Beauville decomposition from the Chow ring of abelian varieties. As a consequence, the rational orbit filtration is opposite to the conjectural Bloch-Beilinson filtration for generalized Kummer varieties. IntroductionThe motivation of this work comes from the study of the Chow group of zero-cycles of a hyper-Kähler manifold. Based on the existence of the Beauville-Voisin zero-cycle in a projective K3 surface [4], which up to a scalar multiple is the intersection of any two divisor classes, Beauville asked in [3] for a projective hyper-Kähler manifold X, whether the Bloch-Beilinson filtration F • BB on the Chow ring of X with rational coefficients CH • (X), if exists, admits a multiplicative splitting as in the case of abelian varieties [2]. While the existence of the Bloch-Beilinson filtration is still largely conjectural, Beauville also formulated his weak splitting conjecture [3] which is predicted by the existence and the splitting of F • BB , but does not rely on the existence of F • BB . The reader is referred to [3,20,8,16] for the partial results of this conjecture. Another way to approach the splitting question, at least for CH 0 (X), is based on the following observation due to Voisin [21, Lemma 2.2]. If C is a curve in a projective K3 surface S such that all points in C are rationally equivalent in S, then the Beauville-Voisin zero-cycle is the class of any point in C. Such a curve and its higher dimensional analogue in a hyper-Kähler manifold led Huybrechts to introduce the following definition : 1 ([11]). -A constant cycle subvariety Y ⊂ X is a subvariety such that all points in Y are rationally equivalent in X.Constant cycle subvarieties are used by Voisin in [22] to introduce the rational orbit filtration S • CH 0 (X) as follows.Definition 1.2 ([22]). -For any integer p, the subgroup S p CH 0 (X) is generated by the classes of points x ∈ X supported on a constant cycle subvariety of dimension p.

We prove that every compact Kähler threefold has arbitrarily small deformations to some projective manifolds, thereby solving the Kodaira problem in dimension 3.

Let X be a compact Kähler manifold of dimension three. We prove that there exists a projective manifold Y such that π 1 (X) ≃ π 1 (Y ). We also prove the bimeromorphic existence of algebraic approximations for compact Kähler manifolds of algebraic dimension dim X −1. Together with the work of Graf and the third author, this settles in particular the bimeromorphic Kodaira problem for compact Kähler threefolds.Date: March 5, 2018.

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