Are Einstein-Hermitian connections on a stable vector bundle ever algebraic? Let $X$ be a smooth, complex projective variety with ample line bundle $H$, and let $E$ be a poly stable vector bundle on $X$. Then there is a unique Hermitian-Einstein connection on $E$. Is this connection ever algebraic?
 A: Some comment: algebraic connection $\nabla$ gives rise to a connection $\nabla+\bar{\partial}$ on the associated holomorphic vector bundle which its curvature vanishes.
A vector bundle which admit algebraic connection has vanishing Chern classes, so if $E$ admits Hermitian-Einstein connection, in general its connection could be non-algebraic and its Chern classes is non-zero in general.
In fact, non-flat algebraic connections for vector bundles not admitting flat structures on complex projective manifolds does not exists.
In general if $E$ be an algebraic vector bundle on a curve with algebraic connection $D$, then E always is semi-stable with degree zero. See this paper of Biswas.
Let $E$ be a semistable vector bundle (which is equivalent with $E$ admits an approximate Hermitian-Einstein structure) on curve $X$ of rank $r$ and degree zero. Then $E$ admits an algebraic connection.
If the holomorphic line bundle $L$ has an algebraic connection, then it is flat. See This answer in MathOverflow.
