Holomorphic vector field

A holomorphic vector field is a holomorphic section of $T^{1,0}X$. In holomorphic coordinates it has the form ${\sum }_j{f}^j(z)\partial /\partial z^j$ with holomorphic coefficient functions $f^j$.

On a complex vector space with linear coordinates, the Euler vector field is $E={\sum }_j{z}_j{\partial }_{z_j}$. It generates scalar dilations, and the Euler homogeneous function theorem says $E(f)=kf$ for a homogeneous function of degree $k$. Contracting a top form with $E$ gives a horizontal form for the quotient by nonzero scalings; descent also requires the appropriate scaling invariance.

A complex-linear endomorphism $A$ of ${\mathbb{C}}^{n+1}$ induces a holomorphic vector field on Complex projective space by differentiating the projective transformations $[z]\mapsto [e^{tA}z]$. Its zeroes are precisely the projective eigenlines of $A$; a diagonalizable $A$ with distinct eigenvalues therefore gives exactly $n+1$ zeroes.