Is positivity relative



Summary:
In this thesis, the geometric properties of the module space of polarized Calabi-Yau manifolds are investigated using methods of complex-analytical differential geometry. For this purpose, families of polarized Calabi-Yau manifolds are considered. The fibers of such a family have unique Ricci-flat Kähler metrics, the cohomology classes of which are given by the polarization. These Kähler metrics induce a Hermite’s metric on the relative canonical bundle of the family whose curvature form is being studied. In addition, a sufficient condition for the existence of a semi-Ricci flat Kähler metric on the total space of a family is shown. Furthermore, certain higher direct image sheaves are considered, which carry natural Hermite’s metrics, which generalize the Weil-Petersson metric on the module space. The curvature tensor of these metrics is calculated and some applications are listed.

Summary:
This thesis deals with geometric properties of the moduli space of polarized Calabi-Yau manifolds employing methods from complex-analytic differential geometry. Given a family of polarized Calabi-Yau manifolds, the fibers possess unique Ricci-flat Kähler metrics with cohomology classes prescribed by the polarization. These Kähler metrics induce a Hermitian metric on the relative canonical bundle of the family, whose curvature form is studied. Moreover, a sufficient condition for the existence of a semi-Ricci-flat Kähler metric on the total space of a family is shown. Furthermore, certain higher direct image sheaves, carrying natural Hermitian metrics which generalize the Weil-Petersson metric on the moduli space, are considered. The curvature tensor of these metrics is calculated and some applications are outlined.