Suppose $X$ is a smooth affine algebraic variety over $\mathbb{C}$ and let $V$ be an algebraic vector field (i.e. an algebraic section of the tangent bundle). If $V$ is locally nilpotent, meaning that for every $f\in\mathbb{C}[X]$ there exists $k\in\mathbb{N}$ such that $V^k(f) = 0$, does it necessarily follow that $V$ is complete? By complete vector field, I mean that the flow $\phi\_t(x)$ of $V$ exists for all $x\in X$ and for all $t\in \mathbb{C}$.



Locally nilpotent algebraic section of tangent bundle is complete?
Yes, it is complete. The flow is given by the operator $e^{tV}$ acting on the coordinate functions $x^i$. This is defined globally because on Zariski open sets the operator is a unipotent linear transformation, so given by polynomials in $t$.


