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Let $X$ be a smooth projective variety over $\mathbb{C}$.

I know that a vector bundle $\mathcal{E}$ on $X$ admits a holomorphic/algebraic connection iff its Atiyah class vanishes, $A(\mathcal{E}) = 0$. If we choose a Hermitian structure on $\mathcal{E}$ giving a Chern connection $\nabla$ then $A(\mathcal{E}) = [\omega_\nabla]$ where $\omega_\nabla$ is the curvature. Therefore, if $\mathcal{E}$ admits a flat Hermitian structure then it admits a holomorphic connection.

I am wondering to what extent this has a converse. Precisely, there are four properties I am interested in:

(1) $\mathcal{E}$ admits a flat connection,

(2) $\mathcal{E}$ admits a flat Hermitian structure,

(3) $\mathcal{E}$ admits a holomorphic connection,

(4) $\mathcal{E}$ admits a flat holomorphic connection.

What are the implications between these properties? We know (2) $\implies (3)$ and obviously (4) $\implies$ (3) and (2) $\implies$ (1) and (4) $\implies$ (1). What about (1) $\implies$ (2) and (3) $\implies$ (4)?

If $\mathcal{E}$ admits a holomorphic connection then we know that $[\omega_\nabla] = 0$ for any Chern connection but I cannot see how to conclude that there exists a flat Chern connection.

I know from How many flat connections has a line bundle in algebraic geometry? that if $\mathcal{E}$ is a line bundle then any holomorphic connection is automatically flat, but it is clear that this is false for rank at least two.

Explicit counterexamples would be helpful.

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    $\begingroup$ Indranil Biswas has various papers on these questions. There does not seem to be one big theorem or paper that handles all of them. $\endgroup$ – Ben McKay Feb 27 at 20:00
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    $\begingroup$ Suppose that (1) holds, then all the Chern classes of $\mathcal{E}$ vanishes. In this case it follows essentially from the Donaldson-Uhlenbeck-Yau Theorem that (2) holds iff $\mathcal{E}$ is $\omega$-polystable (where $\omega$ is any Kähler class on $X$). $\endgroup$ – HYL Feb 28 at 6:16
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In the affine algebraic case there is always an algebraic connection: Let $A$ be a commutative unital ring and let $E$ be a finite rank projective $A$-module. There is the Atiyah sequence

$$ 0 \rightarrow \Omega^1_A \otimes E \rightarrow J^1(E) \rightarrow^{\pi} E \rightarrow 0$$

and since $E$ is projective and $\pi$ surjective, it follows $\pi$ has an $A$-linear split $s: E \rightarrow J^1(E)$.The splitting $s$ corresponds to a connection

$$\nabla: E \rightarrow \Omega^1_A \otimes E$$

and $\nabla$ is "seldom" a flat algebraic connection. There is an explicit formula for the curvature $R_{\nabla}$ of a connection/covariant derivative $\nabla$: Let

$$p: A^n \rightarrow E$$

be a surjection and let $u_1,..,u_n$ be a basis for $F:=A^n$ as $A$-module. Let $s$ be an $A$-linear splitting of $p$

Let $\phi:=s \circ p: F \rightarrow F$ be the idempotent of $E$ corresponding to $p,s$ and let $\phi:=(a_{ij})$ with $a_{ij}\in A$ be the matrix of $\phi$ in the basis $u_i$. Define for any pair $z,x\in \operatorname{Der}(A)$ the matrix

$$L1.\text{ }M:= [(z(a_{ij})), (x(a_{ij}))] \in A^{n \times n}\cong \operatorname{End}_A(F).$$

You may prove that $M $ induce a map $M^*\in \operatorname{End}_A(E)$ and the following formula holds:

Theorem 1: $R_{\nabla}(z,x):= M^* \in \operatorname{End}_A(E)$.

From formula L1 it follows $\nabla$ is seldom a flat connection. Hence in the affine situation it is easy to give explicit examples of non-flat algebraic connections: You must calculate a splitting $s$ of $p$ and the idempotent element $\phi$.

In the projective case it follows $E$ may not have a connection - there is always a cohomology class $a(E)$ which is zero iff $E$ has a connection. If $X$ is a complex projective algebraic manifold, there is a correspondence between flat connections on finite rank vector bundles on $X$ and finite dimensional complex representations of the topological fundamental group $\pi(X)$ of $X$. Given a flat algebraic connection $(E,\nabla)$ on $X$ it follows $E^{\nabla}$ is a local system of finite dimensional complex vector spaces on $X$, and to $E^{\nabla}$ you get a finite dimensional complex representation

$$\rho: \pi(X) \rightarrow GL(V).$$

This correspondence (the "Riemann-Hilbert correspondence") is an "equivalence of categories" in an appropriate sense (this is "vague"). Hence you consider the category of pairs $(E,\nabla)$ where $E$ is a finite rank vector bundle on $X$ with a flat connection $\nabla$, and "maps of connections". You also consider the category of finite dimensional complex representations of $\pi(X)$ and maps of representations. Hence there is "as many" flat connections as representations of the topological fundamental group. This is a well developed theory (originating in some papers of Weil I believe, many people have contributed to this study) from the 40s and 50s. In the below mentioned book you will find this further developed in the framework of "holonomic D-modules" - this is a well developed theory. The book also gives many references.

In the affine situation there is always a space

$$C:=\operatorname{Hom}_A(\operatorname{Der}(A), \operatorname{End}_A(E))$$

of connections, and if $\Omega^1_A$ is a finite rank projective $A$-module you get

$$ C\cong \Omega^1_A \otimes \operatorname{End}_A(E),$$

hence the "parameter-space of connections" is a finite rank vector bundle on $A$. Hence given a connection $\nabla$ with curvature given by Theorem 1, you may add a potential $\phi \in C$ to get a new connection $\overline{\nabla}:=\nabla + \phi$. Hence if you want to study the problem if $E$ has a flat algebraic connection you must study the "moduli space" $C$. Similar for $X$ - this again is a well devleoped theory - "moduli spaces of connections". If you consider the "set of potentials" $\phi \in C$ with the property that the curvature is zero

$$L2.\text{ }R_{\overline{\nabla}}=0$$

you get a subvariety $M^{fl}(E) \subseteq \mathbb{V}(C^*)$ parametrizing flat connections on $E$, and in high dimension and low rank, the system of equations defining $M^{fl}(E)$ will be "overdetermined". Hence it is not clear if $E$ has a flat algebraic connection in general. Hence in the affine algebraic situation there is always a connection which is non-flat in general by formula L1, and it is an open problem to determine if $E$ has a flat algebraic connection. You must study the subvariety you get from equation L2.

Example: If you let $n:=dim(A)\geq 10$ and $rk(E)=2$ you get an overdetermined system of equations defining $M^{fl}(E)$, hence in this case you may get an empty moduli space

$$M^{fl}(E)=\emptyset.$$

For such $E$ you always have a non-flat connection from equation L1 and Theorem 1. The system defining $M^{fl}(E)$ has $\binom{n}{2}$ equations and $rk(\Omega^1\otimes \operatorname{End}(E))=4n$. For $n>>0$ it follows this system does not "have a solution".

Question: "What about (1) ⟹ (2) and (3) ⟹ (4)?"

Answer: I believe it is an old conjecture that if $E$ has an algebraic/holomorphic connection, then $E$ has a flat algebraic /holomorphic connection, but I do not have a precise reference (maybe the paper of Atiyah from 1956 in TrAMS).

You'll find this statement

Citation: "Non-flat algebraic connections for bundles on complex projective manifolds are virtually non-existent (we know of none)"

in the introduction of

https://arxiv.org/pdf/alg-geom/9602001.pdf

Note 1: In Borel's book page 226 you find the following construction: If $X \subseteq \mathbb{P}^n_{\mathbb{C}}$ is a smooth quasi projective algebraic variety and if $\omega^1_X$ is the canonical bundle of $X$, it follows the Lie derivative induce a right $D_X$-module structure on $\omega^1_X$. This does not imply there is a flat algebraic connection

$$\nabla: \omega^1_X \rightarrow \Omega^1_{X}\otimes \omega^1_X.$$

Since $D_X$ is a sheaf of non-commutative rings, there is no obvious relation between left $D_X$-modules and right $D_X$-modules. From a right $D_X$-module $E$ we get a left $D_X$-module $\omega^{-1}_X \otimes E$ and to the canonical bundle $\omega^1_X$ we get the trivial bundle $\mathcal{O}_X$. To a left $D_X$-module we get canonically a flat connection.

Note 2: If $X$ is a complex projective manifold and $E$ is an indecomposable finite rank vector bundle on $X$, it follows $\Gamma(X,\operatorname{End}(E))$ is a finite dimensional algebra over $\mathbb{C}$.

Borel, A. Algebraic D-modules. Perspectives in Mathematics, Vol. 2, Boston etc.: Academic Press, Inc., Harcourt Brace Jovanovich, Publishers. xii, 355 p.; $ 29.95; £25.00 (1987).

PS: I have noticed that some people on this site dislike it when people are editing their posts - is this a big problem?

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    $\begingroup$ Just a remark: the (correct) citation of Bloch-Esnault is very surprising. As soon as you have two 1-forms $\alpha $ and $\beta $ with $\alpha \wedge \beta \neq 0$, you can build a non-flat connection on $\mathscr{O}^2$, by $\nabla e_1=e_2\otimes \alpha $, $\nabla e_2=e_1\otimes \beta $. The serious problem is to find such a connection on a non-flat bundle. $\endgroup$ – abx Mar 1 at 9:51
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    $\begingroup$ P.S.- Editing a post to add some content (as you did here) is fine. The problem is with trivial editing (changing a word...) to have the post put again at the top of the list. $\endgroup$ – abx Mar 1 at 9:55
  • $\begingroup$ @abx - given a connection $\nabla$ you may always add a potential $\phi$ to get a new connection $\nabla^*:=\nabla+ \phi$ and $\nabla^*$ will in general be non-flat. The bundle you consider is the trivial rank 2 bundle, and this bundle has trivially a flat connection. I believe in the above mentioned paper they speak of connections on non-trivial bundles (=not isomorphic to the trivial bundle) - but you should ask the authors of the paper about this. $\endgroup$ – hm2020 Mar 1 at 10:03
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The comment of HYL should be an answer. Since the OP has asked for explicit counterexamples, I will give an example that 1) does not imply 2):

Consider a compact Riemann surface $\Sigma$ of genus $g\geq 2.$ A complex projective structure on $\Sigma$ is given by an atlas of holomorphic coordinates which are related to each other by Moebius transformations (as given by $PSL(2,\mathbb C)$). The developing map $\widetilde\Sigma\to\mathbb CP^1$ is well-defined on the universal covering and induces a $PSL(2,\mathbb C)$ monodromy. It is well-known that (for compact Riemann surfaces) there always exists a lift to a $SL(2,\mathbb C)$ monodromy. The corresponding flat $SL(2,\mathbb C)$-bundle $(V\to\Sigma,\nabla)$ induced from the representation is unstable. In fact, the projective structure can be recovered from $\nabla$ as follows. There is a holomorphic subbundle $S\to V$ such that $$\nabla \colon S\to K_\Sigma V/S$$ is an isomorphism. As $V/S=S$ this implies that $S^2=K_\Sigma.$ Thus, the holomorphic bundle $V$ is unstable. On the other hand, every flat Hermitian bundle must be semi-stable (stable or totally reducible).

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