Saturated orbit closures in the Hodge bundle

We give a new proof of the classification of ${\mathrm{G}\mathrm{L}}^{+}(2,\mathbb{R})$-orbit closures that are saturated for the absolute period foliation of the Hodge bundle. As a consequence, we obtain a short proof of the classification of closures of leaves of the absolute period foliation of the Hodge bundle. Our approach is based on a method for classifying ${\mathrm{G}\mathrm{L}}^{+}(2,\mathbb{R})$-orbit closures using deformations of flat pairs of pants.