Let $B=\lbrace \z\<1\rbrace$ be a unit ball in $\mathbb{C}^n, n\geq 2.$ Let $f_1,\cdots, f_n, f$ be holomorphic functions on $B.$ Now, consider the following first order, linear PDE: $$f_1\partial_{z_1}(y)+\cdots +f_n\partial_{z_n}(y)+fy=0.$$ Here is my question. Is there some nondegeneracy condition of functions $f_1,\cdots, f_n$ that implies existence of a nontrivial solution $y$ which is holomorphic on $B?$ From what I understand about the CauchyKovalevskaya theorem, if $f_1,\cdots, f_n$ have no common zero, then for any $w\in B$ there exists a nontrivial holomorphic solution (in fact infinitely many) defined near $w.$ But, I don't see how to get a global solution from it.Thank you very much in advance.
1 Answer
If the holomorphic vector field $X=\sum_i f_i\partial_{z_i}$ has no zeroes and there is a complex hypersurface nowhere tangent to it, and each point lies on a unique trajectory of that vector field through that hypersurface, then the differential equation becomes a linear ordinary differential equation for a holomorphic parameterization of each trajectory, so has global solvability. So you need to find your nowhere tangent hypersurface, and prove, for each point, both existence and uniqueness of a trajectory through that hypersurface and through that point.

$\begingroup$ I see, thanks! But I don't quite see apriori how doable that is. $\endgroup$– John Z.Sep 7, 2021 at 16:31