= SL2R action on differentials
{c}
{title2=$\mathrm{SL}_2(\mathbb R)\curvearrowright\Omega\mathcal M_g,\mathcal Q\mathcal M_g$}
For nonzero <holomorphic one-forms> and <holomorphic quadratic differentials>, apply $A\in\mathrm{SL}_2(\mathbb R)$ to every <flat coordinate>. Translation and sign transition maps remain of the same type, defining a new <complex structure> and differential. Composition gives a <group action>, <zero orders> persist and area is preserved. The zero section has no such atlas; fixing it gives only a set-theoretic extension that is generally discontinuous.
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