@article { author = {Cao, H.-D. and Sun, X. and Yau, S.-T. and Zhang, Y.}, title = {Weil-Petersson metrics on deformation spaces}, journal = {Journal of the Iranian Mathematical Society}, volume = {1}, number = {1}, pages = {117-128}, year = {2020}, publisher = {Iranian Mathematical Society}, issn = {2717-1612}, eissn = {2717-1612}, doi = {10.30504/jims.2020.104184}, 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 = {‎Weil-Petersson metrics,Deformation Spaces,moduli space}, url = {https://jims.ims.ir/article_104184.html}, eprint = {https://jims.ims.ir/article_104184_b5c880cdd6543fe6c6b09d421a48be6a.pdf} }