What is a Holomorphic Vector Field?

Question

On a smooth manifold $M$, a smooth vector field is an element of $\Gamma(M, TM)$ which is the space of all smooth sections of the bundle $TM \to M$.

If $M$ is a complex manifold, then we have the holomorphic tangent space $T^{1,0}M$. We can form the space $\Gamma(M, T^{1,0}M)$ of smooth sections, but locally, an element can be written as $$f_1\frac{\partial}{\partial z^1} + \dots + f_n\frac{\partial}{\partial z^n}$$ where $n = \mathrm{dim}_{\mathbb{C}}M$ and the functions $f_1, \dots, f_n$ are smooth complex-valued functions, they are not necessarily holomorphic. This makes me think that these vector fields shouldn't be called holomorphic, but maybe I'm wrong.

What is the definition of a holomorphic vector field on a complex manifold?

Any additional resources dealing with such vector fields would also be appreciated.

Answer 1

We can define holomorphic sections of any holomorphic vector bundle in the same way as we define holomorphic functions. Let $X$ be a complex manifold and let $E \to X$ be a holomorphic vector bundle over $X$. We can extend the $\overline\partial$ to act on sections of $E$: Let $E_U \to U \times \mathbb C^r$ be a local trivialization and $(e_1, \dots, e_r)$ be a local holomorphic frame of $E$. If $\sigma = \sum_j s_j e_j$ is a section of $E$ over $U$, then we set $$ \overline\partial \sigma := \sum_j \overline \partial s_j \otimes e_j. $$ If $E_V \to V \times \mathbb C^r$ is another trivialization, then we write $g(z,\lambda) = (z, g(z) \lambda)$ for the induced transition function. These are holomorphic, so $g(z)$ is a $r \times r$ matrix of holomorphic functions. If we write $\sigma_U$ and $\sigma_V$ for the representations of the section $\sigma$ in the frames over $U$ and $V$, then $\sigma_U = g \sigma_V$. It follows that $$ \overline \partial \sigma_U = g \overline \partial \sigma_V $$ because $g$ is holomorphic, so the $\overline \partial$ operator glues to define an operator on the space of sections of $E$.

We now define holomorphic sections of $E$ to be smooth sections $\sigma$ such that $\overline \partial \sigma = 0$. If we pick a local holomorphic frame $(e_1, \dots, e_r)$ and write $\sigma = \sum_j s_j e_j$ as before, then this entails that $\sigma$ is holomorphic if and only if all the functions $s_j$ are holomorphic.

We could of course have defined holomorphic sections as being those sections that satisfy that the "coordinate functions" $s_j$ are holomorphic in any local holomorphic frame. Since the transition functions of $E$ are holomorphic, this is well defined. This is basically the same as what we did here.

Since you ask for additional resources for dealing with holomorphic tangent fields specifially, I encourage you to have a look at the Bochner--Weitzenböck formulas you asked about on MO the other day. These are often used to show that there are no non-zero holomorphic vector fields on a manifold (a fun exercise is to prove this by using the Kähler--Einstein metric on a projective manifold with ample canonical bundle -- try Ballmann or Zheng's books if you need help on this).