On the incompleteness of the Weil-Petersson metric along degenerations of Calabi-Yau manifolds
The classical Weil-Petersson metric on the Teichmüller space of compact Riemann surfaces is a Kähler metric which is complete only in the case of elliptic curves. It has a natural generalization to higher dimensional polarized Kähler-Einstein manifolds and to Calabi-Yau manifolds. There are examples of nodal degenerations of Calabi-Yau 3-folds with finite Weil-Petersson distance. The aim of this note is to characterize such finite distance degenerations, and to describe the possible picture of the completion. By applying Schmid's theory of limiting mixed Hodge structure, the author finds that the center of a degeneration of polarized Hodge structures of weight \(n\) with \(F^n\cong\mathbb{C}\) has finite Weil-Petersson distance if and only if \(NF_\infty ^n=0,\) where \( N\) is the nilpotent monodromy and \(F_\infty ^n\) the limiting filtration. Theorem. The monodromy \(T\) of a nodal degeneration of smooth \(n\)-folds is trivial except possibly in the middle-dimensional cohomology. In the middle-dimension, \(N^2=0\) if \(n\) is odd, and \(T^2=I\) (so \(N=0\)) if \(n\) is even. Theorem. The total space of a small smoothing of terminal singularities has at most terminal singularities. Theorem. Let \(\aleph \prime \Delta \) be a smoothing of a terminal Calabi-Yau 3-fold. Then \(\aleph \prime \Delta \) is not birational to a smooth family.
scientific article; zbMATH DE number 1019794