# When do flat holomorphic connections exist?

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.

• Indranil Biswas has various papers on these questions. There does not seem to be one big theorem or paper that handles all of them. – Ben McKay Feb 27 at 20:00
• 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 $X$). – HYL Feb 28 at 6:16

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.; $29.95; £25.00 (1987). PS: I have noticed that some people on this site dislike it when people are editing their posts - is this a big problem? • Just a remark: the (correct) citation of Bloch-Esnault is very surprising. As soon as you have two 1-forms$\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 $. The serious problem is to find such a connection on a non-flat bundle. – abx Mar 1 at 9:51 • P.S.- Editing a post to add some content (as you did here) is fine. The problem is with trivial editing (changing a word...) to have the post put again at the top of the list. – abx Mar 1 at 9:55 • @abx - given a connection$\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. – hm2020 Mar 1 at 10:03

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).