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 HermitianEinstein connection on $E$. Is this connection ever algebraic?

$\begingroup$ Have you looked at the Chern connection of the KahlerEinstein metric on a manifold with ample canonical bundle? $\endgroup$– Gunnar Þór MagnússonApr 12, 2015 at 19:09

$\begingroup$ I believe that this is essentially the question I am interested in if the tangent bundle is stable. Under what condition is this real analytic connection algebraic? $\endgroup$– Ray HooblerApr 12, 2015 at 21:59
1 Answer
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 HermitianEinstein connection, in general its connection could be nonalgebraic and its Chern classes is nonzero in general.
In fact, nonflat 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 semistable with degree zero. See this paper of Biswas.
Let $E$ be a semistable vector bundle (which is equivalent with $E$ admits an approximate HermitianEinstein 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.

2$\begingroup$ Let $A$ be a special affine variety in the sense of Griffiths. A holomorphic vector bundle $E\to A$ has an algebraic structure if and only if, there exists a Hermitian structure for $E$ whose curvature form satisfies $$\Theta(\sigma)=O(\omega_{Poincare})$$ in the punctured polycylinders at infinity on $A$. See Theorem I. of paper of Griffiths publications.ias.edu/node/210 . So HermitianEinstein metric on special affine varieties are algebraic since $E$ has order $O(1) $ $\endgroup$– user21574Nov 15, 2017 at 9:58

1$\begingroup$ I think you are using the term ``algebraic connection'' to mean a complex algebraic connection on a complex algebraic vector bundle. The question is about real algebraic connections on complex algebraic vector bundles, I think. For example, the FubiniStudy metric is real algebraic, i.e. given in local affine charts by real algebraic functions. $\endgroup$ Apr 10, 2018 at 9:00