# Weil-Petersson metrics on deformation spaces

Document Type : Research Article

Authors

1 Department of Mathematics‎, ‎Lehigh University‎, ‎Bethlehem‎, ‎PA 18015‎, ‎USA.

2 Department of Mathematics, Lehigh University, Bethlehem, PA 18015, USA.

3 Department of Mathematics, Harvard University, Cambridge, MA 02138, USA.

4 Yau mathematical Sciences Center, Tsinghua University, Beijing, 100804, China.

Abstract

In this paper we survey various aspects of the classical wpm and its generalizations‎, ‎in particular on the moduli space of ke manifolds‎. ‎Being a natural $L^2$ metric on the parameter space of a family of complex manifolds (or holomorphic vector bundles) which admit some canonical metrics‎, ‎the wpm is well defined when the automorphism group of each fiber is discrete and the curvature of the wpm can be computed via certain integrals over each fiber‎. ‎We shall discuss the Fano case when these fibers may have continuous automorphism groups‎. ‎We also discuss the relation between the wpm on Teichm"uller spaces of K"ahler-Einstein manifolds of general type and energy of harmonic maps‎.

In this paper we survey various aspects of the classical wpm and its generalizations‎, ‎in particular on the moduli space of ke manifolds‎. ‎Being a natural $L^2$ metric on the parameter space of a family of complex manifolds (or holomorphic vector bundles) which admit some canonical metrics‎, ‎the wpm is well defined when the automorphism group of each fiber is discrete and the curvature of the wpm can be computed via certain integrals over each fiber‎. ‎We shall discuss the Fano case when these fibers may have continuous automorphism groups‎. ‎We also discuss the relation between the wpm on Teichm"uller spaces of K"ahler-Einstein manifolds of general type and energy of harmonic maps‎.

Keywords

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