{title2=$h$}
{wiki=Hermitian_vector_bundle}

A Hermitian metric on a <holomorphic vector bundle> $E\to X$ is a smoothly varying positive-definite <Hermitian form> $h_x$ on every complex fiber $E_x$.


Hermitian metric on a holomorphic vector bundle

The dual of the tangent-normal sequence is the short exact sequence of <holomorphic vector bundle>[holomorphic vector bundles]
$$
0\longrightarrow N_{Y/X}^*\longrightarrow\Omega_X^1|_Y\longrightarrow\Omega_Y^1\longrightarrow0.
$$


Holomorphic conormal sequence

{title2=$N_{Y/X}$}
{wiki=Normal_bundle}

For a <complex submanifold> $Y\subseteq X$, the holomorphic normal bundle is the quotient <holomorphic vector bundle>
$$
N_{Y/X}=TX|_Y/TY.
$$


Holomorphic normal bundle

{wiki}

A holomorphic vector bundle is a complex <vector bundle> whose local trivializations have holomorphic transition functions.


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