Effective positivity of Hodge bundles and applications

Abstract: We show boundedness results that are from different areas of algebraic geometry, but they are closely connected mathematically. The technical starting point is bounding the integer $q>0$ for which the $q$-th Hodge bundles becomes (semi-)positive for families of stable varieties. We then draw many consequences, among which: 1.) a lower bound for the Chow-Mumford volumes of fibrations of fixed dimension and klt general fiber; 2.) a lower bound on the Chow-Moumford volume of moduli spaces of stable varieties, such that at least one parametrized variety is klt, in terms of the dimension and the volume of the parametrized varieties; 3.) a lower bound on the volume of integrable co-rank one lc foliations with klt generic leaf in terms of the dimension; 4.) an upper bound for the cardinality of automorphism groups of fibrations over a curve in terms of dimension and Chow-Mumford volume. We also give pair versions of the results with coefficients varying in a DCC set.