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?

• Have you looked at the Chern connection of the Kahler-Einstein metric on a manifold with ample canonical bundle? Apr 12, 2015 at 19:09
• 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? Apr 12, 2015 at 21:59

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.

• 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 Hermitian-Einstein metric on special affine varieties are algebraic since $E$ has order $O(1)$
– user21574
Nov 15, 2017 at 9:58
• 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 Fubini--Study metric is real algebraic, i.e. given in local affine charts by real algebraic functions. Apr 10, 2018 at 9:00