PomeD4DAF3EEB0E9Lemma 12.30.1.D4DAF3EEB0E9 Choose local inverses $z_ i:U_ i\to \mathbb C$ of the projection $\mathbb C\to E$, and on each component of an overlap write
The cocycle $\{ m_{ij}\vartheta _ E\} $ defines a nonzero class $\eta \in H^1(E,\Omega _ E^1)$, which spans this one-dimensional space.
Proof. Consider the vector bundle obtained from $\mathbb C_ z\times \mathbb C_ a\times \mathbb C_ b$ by the action
The coordinate $b$ gives a map to $\mathcal O_ E$, which defines an extension of $\mathcal O_ E$ by $\Omega _ E^1$, the kernel coordinate $a$ corresponding to $a\vartheta _ E$. The local lifts $s_ i=(z_ i,0,1)$ satisfy $s_ j-s_ i=m_{ij}\vartheta _ E$, as one sees by expressing $s_ i$ in the coordinates over $z_ j$. Thus the extension has class $\eta $.
If it split, the pullback of a holomorphic lift of $1$ to $\mathbb C$ would have the form $(z,g(z),1)$ with $g$ entire and
The derivative $g'$ would be lattice periodic, hence constant by the maximum principle on the compact torus. The first identity then forces this constant to be zero, contradicting the second. Hence $\eta \ne 0$. Serre duality on the compact connected curve gives $H^1(E,\Omega _ E^1)\simeq H^0(E,\mathcal O_ E)^*\simeq \mathbb C$, which proves the spanning assertion. $\square$
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.