I am trying to find the conditions under which a complex circle manifold intersects with a linear subspace. I have a linear subspace of dimension T-K (defined as the null subspace of a $K \times T$ complex matrix) (i.e., a subspace of the space of complex vectors of size $T$, $\mathbb{C}^T$). I have control over $T$ but $K$ is fixed. The complex circle manifold is defined as $\mathcal{C} =\{\mathbf{x} \in \mathbb{C}^T|\; |x_1| = |x_2|=\cdots = |x_T|=1\}$. It has a dimension of $T/2$ as far as I see it (complex dimensions). Both the subspace and the complex circle manifolds are submanifolds of the Euclidean space. I am confused with the concept of dimensions counting (i.e., saying that $(T-K) + (T/2) > T$ would guarantee that the intersection will be non-empty) and I am not sure if it fits here. The reason for my confusion is that if I count the dimensions of two hyperspheres of different radii they will, of course, exceed $T$ dimensions, yet obviously, there is no intersection between them.
Please accept my apology as I am not a math expert and I might be missing something clear.