Study of moduli spaces of projective varieties of general type

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I obtain that there exists a positive number m(n) dependeing only on n such that for every smooth projective n-fold of general type and for every positive integer m> m(n), mK_{X} gives a birtaional embedding of X into a projective space.This is a furthere generalization of the result of E. Bombieri for surfaces of general type.Next I proved that the for a smooth projective family f X longrightarrow S and a semipositive singular hermitian line bundle L,The adjoint bundle K {X/S} + L has a Bergman kernel which is logarithmically plurisubharmonic on $X$.Moreover the curvature current extends to the whole completion as a closed positive current.3rd I have proven a new construction of Kaehler-Einstein metric with negative Ricci curvature. This enables us to study the variation of Kaehler-Einstein volume form on a smooth projective family of canonically polarized varieties.As a consequence I have obtained the plurisubharmonic variation of the Kaehler-Einstein volume form on a smooth projective Family.4^<th>. I have constructed a canonical singular hermitian metric on the canonical bundle of any nonuniruled projective varieties with semipositive curvature current. And minimal singularities (AZD).This enables us to obtain a very short proof of the invariance of plurigenera.