Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
1  
"$f_1\frac{\partial}{\partial z^1} + \dots + f_n\frac{\partial}{\partial z^n}$ where $n = \mathrm{dim}_{\mathbb{C}}M$ and the functions $f_1, \dots, f_n$ are smooth complex-valued functions" Why don't you just replace "smooth" by "holomorphic"? –  Makoto Kato Sep 18 '12 at 8:15
    
I'm not sure whether this is what is meant, and if it is, what the correct global description is (i.e. sections of the bundle $T^{1,0}M \to M$ such that...). –  Michael Albanese Sep 18 '12 at 8:19
1  
The holomorphic tangent bundle has the canonical complex structure. Its holomorphic sections are holomorphic vector fields. The local representation of a holomorphic vector field is of the above form. –  Makoto Kato Sep 18 '12 at 8:30
2  
We can define holomorphic sections of any holomorphic vector bundle in the same way as we define holomorphic functions: if $E \to X$ is a bundle, then $\overline \partial$ acts on $E$ (define it locally as usual and observe that the resulting operator glues b/c the transition functions are holomorphic). Then a holomorphic section $\sigma$ of $E$ is a smooth section such that $\overline \partial \sigma = 0$. In particular, this entails that your local functions $f_j$ are holomorphic. –  Gunnar Magnusson Sep 18 '12 at 8:32
    
Would either/both of you be willing to write your comment in an answer so that I may accept it? –  Michael Albanese Sep 18 '12 at 8:35

1 Answer 1

up vote 3 down vote accepted

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).

share|improve this answer
    
Thanks for your help yet again. –  Michael Albanese Sep 18 '12 at 13:59
2  
"We are here to help each other get through this thing, whatever it is." –  Gunnar Magnusson Sep 18 '12 at 14:52

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.