Additivity of Kodaira dimension and hyperbolicity for families of varieties

Abstract

To characterize morphisms between complex algebraic varieties, we prove a series of results on the behavior of positivity, hyperbolicity, and Kodaira dimension under morphisms of quasi-projective varieties. In the first part of the dissertation, we prove a logarithmic base change theorem for pushforwards of pluri-canonical bundles and use it to deduce that positivity properties of log canonical divisors descend via smooth projective morphisms. As a consequence, we prove Popa's conjecture on the superadditivity of the logarithmic Kodaira dimension of smooth algebraic fiber spaces over bases of dimension at most three and analyze related problems. In the second part, we prove the analogue of Viehweg's hyperbolicity conjecture for Whitney equisingular families of projective varieties with Gorenstein rational singularities whose geometric generic fiber has a good minimal model. Namely, for such families with maximal variation, the base spaces are of log general type. This construction suggests an equisingular stratification of the moduli space of varieties of general type, with each stratum being hyperbolic, and our result is a first step in this direction.