Let $\omega$ be a holomorphic $1$-form on a complex curve $C$ of genus $g$. I want to make sure that I have the correct notion of the period of a holomorphic $1$-form in terms of the period matrix in mind.

I am a bit confused about this because in Griffiths and Harris he defines periods as the columns of the period matrix but I can't find a definition of the period of an arbitrary $1$-form $\omega$.

I think the period of $\omega$ could be described by the following row vector

$\begin{bmatrix} \int_{\gamma_1} \omega & \int_{\gamma_2} \omega & \cdots & \int_{\gamma_{2g}} \omega \end{bmatrix} $

where $\gamma_i$ are a basis of $H^1(C, \mathbb{Z})$. Now suppose that $\omega_1, \dots, \omega_g$ is a basis for $H^0(C, \Omega^1)$ so that

$\omega= m_1 \omega_1 + \dots + m_g \omega_g$ where $m_i \in \mathbb{C}$.

Then I think we could write the period of $\omega$ in terms of the period matrix with some extra coefficients, i.e,

$\begin{bmatrix} m_1\int_{\gamma_1} \omega_1 & m_1 \int_{\gamma_2} \omega_1 & \cdots & m_1 \int_{\gamma_{2g}} \omega_1 \\ m_2\int_{\gamma_1} \omega_2 & m_2 \int_{\gamma_2} \omega_2 & \cdots & m_2 \int_{\gamma_{2g}} \omega_2 \\ \vdots \\ m_g\int_{\gamma_1} \omega_g & m_g \int_{\gamma_2} \omega_g & \cdots & m_g \int_{\gamma_{2g}} \omega_g \end{bmatrix} $

and then define the period of $\omega$ to be the sum of the column vector of the above matrix.

  1. Is my definition of the period of $\omega$ correct?

Now let $\Lambda$ be the lattice formed by the columns of the period matrix which are known to be linearly independent via the Riemann bilinear relations.

It would be great if elements of this lattice are periods of holomorphic $1$ forms on $C$. However, the lattice is defined over $\mathbb{Z}$ and unfortunately the coefficients in the above matrix are in $\mathbb{C}$.

Are periods of arbitrary $\omega$ not elements of the lattice $\Lambda$ ?


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