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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
  • $\begingroup$ Any flat (complex linear) connection is automatically holomorphic because there are transition functions which are constant (complex linear) matrices, so (1) and (4) are equivalent. $\endgroup$ – complex May 26 at 8:12
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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

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).

Note 3: There is the following general result: If $(L,a)$ is a Lie-Rinehart algebra and $(E,\nabla)$ is a connection, there is a non-abelian extension (the non-abelian Atiyah extension)

$$N1.\text{ } 0 \rightarrow \operatorname{End}_A(E) \rightarrow L(E,\nabla) \rightarrow L \rightarrow 0 $$

of Lie-Rinehart algebras which splits iff $E$ has a flat connection. There is a "cohomology set" $\operatorname{Ext}^1(L,(E,\nabla))$ classifying such non-abelian extensions. By definition

$$L(E,\nabla):=\operatorname{End}_A(E)\oplus L$$

with the following Lie product:

$$[(\phi,x),(\psi,y)]:=([\nabla(x),\psi]-[\nabla(y), \phi]+R_{\nabla}(x,y),[x,y])$$

for all $\phi, \psi\in \operatorname{End}_A(E)$ and $x,y\in L$. You must check that the sequence $N1$ splits iff there is an $A$-linear map $P: L \rightarrow \operatorname{End}_A(E)$ such that $\overline{\nabla}:=\nabla+P$ is a flat connection. A similar construction exists globally.

Hence given a holomorphic vector bundle $E$ on a complex projective manifold $X$, there are two obstructions: The Atiyah class $a(E)$ which is zero iff $E$ has a holomorphic (or algebraic) connection. The extension class $n(E)$ given by sequence $N1$ is zero iff $E$ has a flat holomorphic connection. The sequence $N1$ exists whenever $E$ has a holomorphic (or algebraic) connection.

Example: If $\mathbb{P}^n_k$ is complex projective space and if $\mathcal{O}(d)$ is the invertible sheaf with $d \geq 1$, it follows the Atiyah sequence

$$0 \rightarrow \Omega^1 \otimes \mathcal{O}(d) \rightarrow J^1(\mathcal{O}(d)) \rightarrow \mathcal{O}(d) \rightarrow 0$$

does not split, hence $\mathcal{O}(d)$ does not have a holomorphic (or algebraic) connection.

Any finite rank projective $A$-module $E$ with a non-zero class in $\operatorname{H}^2_{DR}(A)$ will have an algebraic connection and no flat algebraic connection. Hence the Atiyah sequence splits, but the sequence $N1$ does not split. I belive there is an explicit example in Loday's book "Cyclic Homology".

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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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    $\begingroup$ I took the liberty of replacing the direct-to-pdf link of an unspecified arXiv preprint with a full bibliographic reference, a link to the arXiv abstract and a stable doi link to the published version. This will help people reading the answer to know whether they should check out the paper or not, especially if they know of it already. $\endgroup$ – David Roberts Jun 21 at 3:50
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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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