Is positivity relative
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.
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.
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