I am reading the paper [1], which states
Haupt showed that a vector with complex entries $(w_1, \cdots, w_g, z_1, \cdots, z_g)$ is the period row of some holomorphic differential with respect to a canonical homology basis on some closed Riemann surface of genus $g$ if and only if
(a) $i\sum_j (w_j\bar{z_j} - \bar{w_j}z_j) > 0$
and
(b) no transformation corresponding to a change of canonical basis will bring the vector to the form $(w,0,\cdots,0,z,0,\cdots,0)$
I am looking at the necessity of these conditions for the time being. Condition (a) is standard and is covered in standard textbooks. However, I do not see any mention of part (b) in these textbooks, nor am I able to see why (b) should hold.
The only related thing I can think of are conditions on $g \times 2g$ period matrices which state that when the left $g \times g$ block of a period matrix is identity then the right $g \times g$ block is symmetric and has positive definite imaginary part. This seems to allow period vectors of the form specified in (b).
Why does (b) hold and how does one interpret this condition?
