Lemma 12.27.1.030B497D5253 Let $D$ be an elliptic curve with a flat Hermitian metric, let $\vartheta $ be a parallel $(1,0)$-form of unit length, and let $v$ be its dual vector field. Let $L$ be a unitary flat line bundle on $D$, with flat connection $\nabla =\partial _ L+\bar\partial _ L$. For every smooth $L$-valued $(0,1)$-form $u$, there is a unique smooth section $f$ of $L$, orthogonal to the parallel sections, such that
\[ \bar\partial _ L f=\iota _ v\partial _ Lu. \]
If $\Pi _ L^{0,1}$ denotes the orthogonal projection onto the parallel $L$-valued $(0,1)$-forms, then, for every Sobolev norm $H^ k$ defined by the flat structure,
\[ \| f\| _{H^ k}=\| (I-\Pi _ L^{0,1})u\| _{H^ k} \leq \| u\| _{H^ k}. \]
In particular, the bound does not depend on the holonomy of $L$.
Proof.
Put $P=\iota _ v\partial _ L$. Since the metric and $L$ are flat, $P=\nabla _ v$ and $P^*=-\nabla _{\bar v}$, and the Laplacian $\Delta _ L$ satisfies $\Delta _ L=P^*P=PP^*$ on sections and on $(0,1)$-forms. In degree $q\in \{ 0,1\} $, let $G_ q$ be the inverse of $\Delta _ L$ on the orthogonal complement of its kernel, extended by zero on the kernel. The operators $P$ and $P^*$ commute with $G_ q$. Set
\[ f=\bar\partial _ L^*G_1Pu. \]
The form $Pu$ is orthogonal to the parallel forms, so $\bar\partial _ Lf=Pu$; also $f$ is orthogonal to the parallel sections. Uniqueness follows because the kernel of $\bar\partial _ L$ on sections consists of parallel sections. Moreover,
\[ \| f\| _{L^2}^2 =\langle Pu,G_1Pu\rangle =\langle u,(I-\Pi _ L^{0,1})u\rangle . \]
Covariant differentiation along a global parallel real frame commutes with all these operators, so applying the identity to every derivative proves the $H^ k$ assertion. The argument never divides by a lower bound for a nonzero eigenvalue of $\Delta _ L$: the derivative in $Pu$ compensates for small eigenvalues.
$\square$
Pome
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