Pome68FEC9C7A0BCLemma 12.30.2.68FEC9C7A0BC Every holomorphic line bundle $A$ of positive degree $d$ on $E$ is defined by factors of automorphy of the form
where $\chi :\Lambda \to \mathrm U(1)$ is a character. In particular,
Proof. Let $A_ d$ be the $d$th tensor power of the degree-one bundle of example 12.29.2. Its factors are the stated exponential factors, and its degree is $d$. The bundle $F=A\otimes A_ d^*$ has degree zero. Start with a smooth Hermitian metric on $F$; its Chern curvature has integral zero. Solving the scalar Poisson equation on the flat compact torus gives a smooth real function whose addition to the weight of the metric makes the curvature vanish. The resulting Chern connection is flat and unitary.
On the simply connected cover $\mathbb C$, parallel transport gives a global parallel frame of unit length of the pullback of $F$. It is holomorphic, because the $(0,1)$-part of the Chern connection is the $\bar\partial $-operator of $F$. Each lattice element changes this frame by a constant unitary factor, and these factors form a character. Tensoring with the given frame of $A_ d$ gives factors of the stated form for $A$, and differentiation gives the last formula. $\square$
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