Mathematics

I. Introduction II. Construction of a completion of the image of a period mapping III. Curvature properties of the extended Hodge line bundle (A) IV. Curvature properties of the extended Hodge line bundle (B) V. Proof that the extended Hodge line bundle is ample VI. Curvature properties of the Hodge vector bundle

In algebraic geometry the use of complex analytic methods differential geometry and PDEs | is long standing and far reaching for rather deep reasons the results these methods give (Hodge theory, vanishing theorems) have not been replaced by purely algebraic techniques. Two central areas in the subject are moduli and classification of varieties today we will discuss two results in which the use of analytic methods through the curvature properties of the Hodge bundles plays a central role.