Hermitian metric that induces a holomorphic splitting

Question

Let $E\rightarrow X$ be a holomorphic vector bundle and $S\hookrightarrow E$ a holomorphic subbundle. Can we always pick a Hermitian metric $h$ on $E$ such that $S^{\perp}$ is holomorphic and the decomposition $$E\simeq S\oplus S^{\perp}$$ holomorphic? If not what would be the obstruction?

Answer 1

Here is a question addressing this. When does a SES of vector bundles split?

Morally, you should not expect holomorphic splittings to come from a Hermitian metric. Indeed, you can't require your metric $h$ to be holomorphic because global sections of holomorphic bundles are hard to come by in general. So, you shouldn't expect to get more than smooth splittings by these metric techniques.

Answer 2

In general we have the following:

Proposition

Let $(E,h)$ be a holomorphic Hermitian vector bundle over a complex manifold $X$ and let $\nabla$ its Chern connection (the unique connection on $E$ which is compatible with the metric $h$ and $\nabla^{0,1}=\bar{\partial}$). Let $S\subset E$ be a smooth complex vector subbundle which is invariant by $\nabla$, i.e. $\nabla$ maps sections of $S$ in sections of $S\otimes T^*X$. Consider $S^\bot$ its orthogonal complement with respect to $h$ (which is a smooth complex vector bundle). Then $S,S^{\bot}$ are both holomorphic complex vector bundle and $E$ splits holomorphically as: $$E\cong S\oplus S^{\bot}$$

The proof is very simple. Infact one has only to show that if $\sigma$ is a local holomorphic section of $E$ then, in according with the $C^{\infty}$ decomposition $\sigma=\sigma_S+\sigma_{S^{\bot}}$, it follows that both $\sigma_S$ and $\sigma_{S^{\bot}}$ are local holomorphic sections of $S$ and $S^{\bot}$ respectively.

Using this proposition one can find an obstruction to the existence of an holomorphic structure on the orthogonal vector bundle $S^{\bot}$ starting from a holomorphic subbundle $S\subset E$. Let's see how:

Let $(E,h)$ be the holomorphic hermitian vector bundle and $S\subset E$ a holomorphic subbundle. Then, as you say, we have a $C^{\infty}$ decomposition $E\cong_{C^\infty}S\oplus S^{\bot}$. Let $\nabla$ be the Chern connection on $E$ and we define $\nabla_S$ and $\alpha$ as $$\nabla(\sigma)=\nabla_S(\sigma)+\alpha\cdot\sigma, \qquad \sigma\in\Omega^0(X;S)$$ where $\nabla_S(\sigma)\in\Omega^1(X;S)$ and $\alpha\cdot\sigma\in\Omega^1(X;S^{\bot})$. Then one can prove:

Proposition

1. $\nabla_S$ is the Chern connection of $(S,h_S)$, where $h_S$ is the restriction of the metric $h$ to $S$.
2. $\alpha$ is a $(1,0)$-form with value in Hom$(S,S^{\bot})$, i.e. $\alpha\in\Omega^{1,0}(X;Hom(S,S^{\bot}))$. Moreover such $\alpha$ is called the second fundamental form.

In according with the two propositions above and the formula $\nabla(\sigma)=\nabla_S(\sigma)+\alpha\cdot\sigma$ it follows that:

Corollary

If $\alpha$ vanishes identically, then $S^{\bot}$ is a holomorphic vector bundle and $E$ splits holomorphically as $$E\cong S\oplus S^{\bot}$$

I hope this will help. The reference is the book "Differential geometry of complex vector bundle" written by Kobayashi.