Fix an integer $ m $ and a lattice $ \Lambda \subset \mathbb{C}^{m} $. Identify $ \Lambda \otimes \mathbb{R} $ with $ \mathbb{C}^{m} $.
Take $ n $ holomorphic functions $ f_{1}, \ldots, f_{n}: U \to \mathbb{C}^{m} $, $ U $ open in $ \mathbb{C}^{k} $. Let $ p \in U $, call $ V_{p} $ the smallest $ \mathbb{R}$-subspace of $ \mathbb{R}^{2m} = \mathbb{C}^{m} $ with a basis in $ \Lambda $ containing $ f_{1}(p),\ldots, f_{n}(p) $ and let $ d_{p} = \dim (V_{p})$.
Fix a point $ p_{0} \in U $ such that $ V_{p_{0}} $ has maximal dimension.
Is it possible to say anything about the set of points $ p\in U $ such that $ V_{p} = V_{p_{0}}$? What about the set of $ p\in U $ such that $ d_{p} = d_{p_{0}}$?
