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Question: is there a simple description of holomorphic vector fields on the complex projective space $\mathbb{C P}^n$ ?

More precisely : for $n=1$, the holomorphic vector fields are of the form $P(z) \frac{d}{dz}$ in coordinates, where $P$ is a polynomial of degree less than 2. Is there a similar description in terms of homogeneous coordinates for $n >1$ ?

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You can describe them using the Euler sequence. It turns out that they are the classes of homogeneous linear polynomial vector fields on $\mathbb{C}^{n+1}$ $$ \sum A_i(z_0, \dots , z_n)\frac{\partial}{\partial z_i} $$ ($A_i$ homogeneous polynomial of degree one) modulo $$R = \sum z_i \frac{\partial}{\partial z_i} $$ the radial vector field.

Take a look at Huybrechts' book Complex Geometry, an introduction.

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