CALABI-YAU MANIFOLDS OF TYPE A AND THE CONE CONJECTURE FOR HYPERELLIPTIC VARIETIES

A Calabi-Yau n-fold Y is a compact, complex, Kähler n-fold with trivial canonical bundle and h^{i,0}(Y)=0 for 0<1 and Oguiso and Sakurai classified them in dimension 3. In this thesis we provide higher dimensional examples that ensure the existence of Calabi-Yau manifolds of type A in all odd dimension. After that, we study the geometry of Calabi-Yau 3-folds of type A. Thanks to Oguiso and Sakurai, we know that there exist only two families F^A_G of Calabi-Yau 3-folds A/G of type A and each of them corresponds uniquely to a group G which acts freely on A and does not contain any translations. More in details, the family F^A_{D_4} is constructed by Catanese and Demleitner and the family F^A_{(Z/2Z)^2} is constructed here. In particular, these families are irreducible and each X in F^A_G admits a finite étale cover A which splits into the product of three elliptic curves. Our main results include the full classification of the automorphisms group and the possible quotients of manifolds in F^A_G for both choices of G. We prove that Aut(X) is finite for X in F^A_G. Furthermore, if X in F^A_{D_4} then X/U is birational to a Calabi-Yau 3-folds for every U in Aut(X), while if X in F^A_{(Z/2Z)^2} then X/\U is birational either to a Calabi-Yau 3-fold or to a 3-fold with negative Kodaira dimension or to a 3-fold Y with trivial Kodaira dimension and K_Y is not 0. In particular, we compute the Hodge numbers and the fundamental groups of Calabi-Yau 3-folds Y birational to the quotient X/U where X in F^A_G and U in Aut(X). In the second part, we establish the Morrison-Kawamata cone conjecture for hyperelliptic varieties. The cone conjecture, an open problem in birational geometry, predicts that the nef and movable cones, of projective varieties Y with K_Y numerically trivial, admit a rational polyhedral structure under the action of automorphism and birational automorphism groups, respectively. The conjecture provides insights into both the geometry and birational geometry of varieties, specially from the point of view of Minimal Model Program. To prove the conjecture for hyperelliptic varieties, we generalize the techniques established by Prendergast-Smith for abelian varieties. Additionally, we investigate whether hyperelliptic varieties A/G have a rational polyhedral nef cones. We obtain a result in relation to the representation of G, in particular we deduce that all the Calabi-Yau manifolds of type A studied in this thesis have rational polyhedral nef cones. Finally, we study the extremal rays of the nef cones of these latter manifolds. We prove that the extremal rays are semi-ample divisors, by generalizing a proof of Oguiso and Sakurai. Thereby, we deduce that all nef divisors are semi-ample divisors.