Weil-Petersson metrics on deformation spaces

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.

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