Let $X=V/\Gamma$ be a <complex torus>. A <Riemann form on a complex torus> is a <Hermitian form> $H$ on $V$ whose imaginary part
$$
E(v,w)=\operatorname{Im}H(v,w)
$$
takes integer values on $\Gamma\times\Gamma$. Equivalently, $E$ is an integral alternating form satisfying
$$
E(iv,iw)=E(v,w),
\qquad E(iv,v)>0
$$
for every nonzero $v$. A <polarisation of a complex torus> is such a positive Riemann form, or equivalently its integral cohomology class. By the <Appell–Humbert theorem>, it is the first Chern class of an ample holomorphic line bundle. A polarisation is principal when the homomorphism $\Gamma\to\Gamma^*$ induced by $E$ is an isomorphism.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 126 / 1 / i / Solution

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{title2=$H$}

A Riemann form on $V/\Gamma$ is a Hermitian form $H$ whose imaginary part is integer-valued on $\Gamma\times\Gamma$. Its imaginary part is equivalently an integral alternating form compatible with the complex structure.


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 Incoming links: Riemann form on a complex torus