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The main goal of this work is to construct and study a reasonablecompactification of the strata of the moduli space of Abelian differentials.This allows us to compute the Kodaira dimension of some strata of the modulispace of Abelian differentials. The main ingredients to study thecompactifications of the strata are a version of the plumbing cylinderconstruction for differential forms and an extension of the parity of theconnected components of the strata to the differentials on curves of compacttype. We study in detail the compactifications of the hyperelliptic minimalstrata and of the odd minimal stratum in genus three.

The Deligne-Mumford and the Incidence Variety Compactifications of the Strata of $\Omega\mathcal{M}_{g}$