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Consider an $n$-dimensional complex manifold $M\subset\mathbb{C}^N$ and let

$$f:\mathcal{U}\subset\mathbb{C}^n\rightarrow \mathcal{V}\subset M\subset\mathbb{C}^N$$

be a local parametrization of $M$.

Assume that for all $p\in\mathcal{U}$ we have that $f(p)$ is a linear combination of $\frac{\partial f}{\partial x_1}(p),\dots, \frac{\partial f}{\partial x_n}(p)$.

Is then $M\subset\mathbb{C}^N$ an affine subspace of $\mathbb{C}^N$?

In general, does there exist a term for a point $q = f(p)\in M$ such that $f(p)$ is a linear combination of $\frac{\partial f}{\partial x_1}(p),\dots, \frac{\partial f}{\partial x_n}(p)$?

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The answer to the first question is No.

The assumption you made is equivalent to stating that for every $q\in M$ that the vector $q\in T_qM$.

This is satisfied whenever $M$ is a portion of a cone, which need not be affine.

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  • $\begingroup$ You are right. Is there something else besides affine spaces and portions of cones? $\endgroup$
    – LaGra
    May 18 at 21:40

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