CALABI-YAU MANIFOLDS OF TYPE A AND THE CONE CONJECTURE FOR HYPERELLIPTIC VARIETIES
type: CreativeWork
description: 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 v…
abstract: 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 v…
author: MONTI, MARTINA
language: eng
keywords: Settore MAT/03 - Geometria, Settore MATH-02/B - Geometria
date published: 2024-12-05
date modified: 2026-07-27T20:18:53Z
provider: Università degli Studi di Milano
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: Università degli Studi di Milano

PropertyValue
property ID: Handle
value: https://hdl.handle.net/2434/1119759

PropertyValue
property ID: DOI
value: 10.13130/monti-martina_phd2024-12-05