On the incompleteness of the Weil-Petersson metric along degenerations of Calabi-Yau manifolds
type: Item
instance of: scholarly article
title: On the incompleteness of the Weil-Petersson metric along degenerations of Calabi-Yau manifolds
zbMATH Open document ID: 0881.32017
author: Chin-Lung Wang
published in: Mathematical Research Letters
publication date: 1998-01-27T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 32S35
Mathematics Subject Classification ID: 14J32
Mathematics Subject Classification ID: 32J17
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 1019794
zbMATH Keywords: Weil-Petersson metric
zbMATH Keywords: degenerations of Calabi-Yau manifold
zbMATH Keywords: mixed Hodge structure
reviewed by: V. V. Chueshev
MaRDI profile type: Publication
DOI: 10.4310/MRL.1997.V4.N1.A14
description: scientific article; zbMATH DE number 1019794