Obstruction theory and the level n elliptic genus

Abstract

Given a height at most two Landweber exact $\mathbb {E}_\infty$-ring $E$ whose homotopy is concentrated in even degrees, we show that any complex orientation of $E$ which satisfies the Ando criterion admits a unique lift to an $\mathbb {E}_\infty$-complex orientation $\mathrm {MU} \to E$. As a consequence, we give a short proof that the level $n$ elliptic genus lifts uniquely to an $\mathbb {E}_\infty$-complex orientation $\mathrm {MU} \to \mathrm {tmf}_1 (n)$ for all $n\, {\geq}\, 2$.