Effective positivity of Hodge bundles and applications

We prove new boundedness results across different areas of algebraic geometry, stemming from a unifying technical starting point: bounding the integer $q > 0$ such that the $q$-th Hodge bundle becomes (semi-)positive for families of stable varieties. This result allows us to show that for stable families $f: X \to T$ of maximal variation with klt general fiber and relative dimension $n$ there exist the following bounds: 1) a lower bound for the Chow-Mumford volume $\left( \lambda_{CM,f} \right)^{\dim T}$ of the form $\delta^{\dim T}$, where $\delta$ is uniform; 2) a uniform lower bound on $K_{X/T}^{n+1}$, when $T$ is a curve; 3) an upper bound for $|\mathrm{Aut}(f)|$ when $T$ is a curve, depending uniformly linearly on $K_{X/T}^{n+1}$. Additionally, we draw several several consequences on the subspaces of the moduli space of stable varieties parametrizing at least one klt variety, such as the positivity of Hodge bundles and a lower bound on the Chow-Mumford volume in terms of the dimension and the volume of the parametrized varieties (the volume is needed only if working on the coarse moduli space). We also give pair versions of the above results with coefficients varying in a DCC set.