On the Kodaira dimension of moduli spaces of Abelian differentials

This paper lays the foundation for determining the Kodaira dimension of the projectivized strata of Abelian differentials with prescribed zero and pole orders in large genus. We work with the moduli space of multi-scale differentials constructed in [BCGGM2] which provides an orbifold compactification of these strata. We establish the projectivity of the moduli space of multi-scale differentials, describe the locus of canonical singularities, and compute a series of effective divisor classes. Moreover, we exhibit a perturbation of the canonical class which allows the corresponding pluri-canonical forms to extend over the locus of non-canonical singularities. As applications, we certify general type for strata with few zeros as well as for strata with equidistributed zero orders when g is sufficiently large. In particular, we show general type for the odd spin components of the minimal strata for g > 12.