# How many flat connections has a line bundle in algebraic geometry?

Suppose $X$ is a projective variety over $\mathbb C$. I am happy to entertain more or different adjectives — I'm not looking for the most general statement, but rather to understand when and how smooth-manifold intuition leads me astray. I know very little algebraic geometry, and so please forgive and correct me if a statement below is mistaken.

It is rare for a line bundle $\mathcal L \to X$ to have a nowhere-vanishing section, and when it does, there are usually very few (only $\mathbb C^\times$ many). Suppose instead that I ask for a weaker structure than for $\mathcal L$ to have a section, but rather let me ask only that it has a flat connection. My question is:

In algebraic geometry, how often does a line bundle have a flat connection? When it has a flat connection, how many flat connections can it have?

I presume that your variety $X$ is smooth.

Consider the additive map $\mathrm d\log \colon \mathscr O_X^*\to \Omega^1_X$ that sends $f$ to $\mathrm df/f$. It induces a map $c_1$ in cohomology from $H^1(X,\mathscr O_X^*)$ to $H^1(X,\Omega^1_X)$ — a coherent avatar of the first Chern class. By Hodge Theory, $H^1(X,\Omega^1_X)$ is a subspace of $H^2(X,\mathbf C)$ and the two notions of first Chern class coincide.

A line bundle $\mathscr L$ has a connection if and only if its first Chern class $c_1(\mathscr L)\in H^1(X,\Omega^1_X)$ vanishes. The proof is straightforward: take an open cover $(U_i)$ of $X$, an invertible section $s_i$ of $\mathscr L$ on $U_i$ and the associated cocycle $(f_{ij})$ representing your line bundle in $H^1(X,\mathscr O_X^*)$. A connection $\nabla$ maps $s_i$ to $s_i\otimes\omega_i$, for some 1-form $\omega_i\in H^0(U_i,\Omega^1_X)$. The condition that these $s_i\otimes\omega_i$ come from a global connection on $X$ is exactly the vanishing of $c_1(\mathscr L)$.

It is a non-trivial fact that if $\mathscr L$ has an algebraic connection, then it is automatically flat. Torsten Ekedahl gave an algebraic proof on this thread of MO (Ekedahl also observes that $p$th power of line bundles in characteristic $p$ have an integrable connection), but an analytic proof seems easy. The algebraic connexion $\nabla$ gives rise to a connexion $\nabla+\bar\partial$ on the associated holomorphic line bundle. One checks that the curvature of this connection is a $(2,0)$-form, while it should be a $(1,1)$-form. Consequently, it vanishes.

When non empty, the set of flat connections on a vector bundle $\mathscr E$ is an affine space under $H^0(X,\Omega^1_X\otimes\mathscr E\mathit{nd}(\mathscr E))$, a finite dimensional vector space. In our case, $\mathscr L$ is a line bundle, hence $\mathscr E\mathit{nd}(\mathscr L)$ is the trivial line bundle so that we get $H^0(X,\Omega^1_X)$.

NB. Following the comment of Ben McKay, I edited the last paragraph.

• Isn't $End(L)$ trivial, since $L$ is a line bundle? – Ben McKay Mar 8 '13 at 15:09
• @Ben McKay. You're absolutely right... I'll edit the answer. – ACL Mar 9 '13 at 0:48
• A priori, there is no reason for the connection $\nabla +\overline{partial}$ to have curvature of type (1,1) (and hence to be flat). It is the curvature of the Chern connection that satisfies a reality property (if you have fixed an hermitian structure). Also, you mean to write that $\nabla+\overline{\partial}$ is a connection on the complex line bundle corresponding to $\mathscr{L}$, not the holomorphic one. – Peter Dalakov Mar 10 '13 at 19:37
• You do have though that all of the Chern classes vanish: the characteristic ring is generated by the Atiyah class, which is zero: see Thm 4 in Atiyah's paper. – Peter Dalakov Mar 10 '13 at 19:52
• @ACL - could you give a reference to a proof that $H^0(X,\Omega^1_X\otimes\mathscr E\mathit{nd}(\mathscr E))$ is finite dimensional? – hm2020 May 26 at 14:25

A vector bundle with an algebraic connection has to have vanishing all Chern classes, at least in characteristic zero. I remember that this follows from the vanishing of the "Atiyah class", but I don't know the details.

• @ Piotr Achinger Yes, you are right. However, I can't find the detail on any book. Hence, I write the detail by myself. If you like, I can sent it to you. – swalker Nov 25 '16 at 3:39

Question: "In algebraic geometry, how often does a line bundle have a flat connection? When it has a flat connection, how many flat connections can it have?"

Comment: Over other fields/rings than the complex number field you get similar results using basic properties of the $$\operatorname{Ext}$$ and $$\operatorname{Hom}$$ functors: If $$\pi: X \rightarrow S:=Spec(A)$$ is any scheme and if $$L\in Pic(X)$$ is any invertible sheaf, there is the Atiyah-sequence

$$A1.\text{ } 0 \rightarrow \Omega^1_{X/S} \otimes L \rightarrow J^1(L) \rightarrow L \rightarrow 0$$

which is split iff $$L$$ has a connection. You get an extension class

$$a(L) \in \operatorname{Ext}^1_{\mathcal{O}_X}(L, L\otimes \Omega^1_{X/S}) \cong$$

$$\operatorname{Ext}^1_{\mathcal{O}_X}(\mathcal{O}_X, L^*\otimes L\otimes \Omega^1_{X/S}) \cong \operatorname{Ext}^1_{\mathcal{O}_X}( \mathcal{O}_X, \Omega^1_{X/S}) \cong$$

$$\operatorname{H}^1(X, \Omega^1_{X/S}).$$

You may view the image of the class $$a(L)=c_1(L)$$ under the above isomorphisms as the first Chern class of $$L$$. Hence $$c_1(L)=0$$ iff $$L$$ has a connection. The set of connections on $$L$$ is parametrized by the set

$$\operatorname{Hom}_{\mathcal{O}_X}(L, L \otimes \Omega^1_{X/S}) \cong$$

$$\operatorname{Hom}_{\mathcal{O}_X}(\mathcal{O}_X,\operatorname{Hom}_{\mathcal{O}_X}(L, L \otimes \Omega^1_{X/S})) \cong$$

$$\operatorname{Hom}_{\mathcal{O}_X}(\mathcal{O}_X, L^*\otimes L \otimes \Omega^1_{X/S}) \cong$$

$$\operatorname{Hom}_{\mathcal{O}_X}(\mathcal{O}_X, \Omega^1_{X/S}) \cong \operatorname{H}^0(X, \Omega^1_{X/S}).$$

More generally if $$E$$ is any locally trivial finite rank sheaf you get a class

$$a(E) \in \operatorname{H}^1(X, \operatorname{End}_{\mathcal{O}_X}(E) \otimes \Omega^1_{X/S}).$$

with the same properties. The "parameter-space of connections" is the set

$$\operatorname{H}^0(X, \operatorname{End}_{\mathcal{O}_X}(E) \otimes \Omega^1_{X/S}).$$

Remark: You may let $$S$$ be any base-scheme and the above results still hold.

The following link gives an obstruction sequence for a general $$E$$ to have a flat connection using non-abelian extensions of (sheaves of) Lie-Rinehart algebras:

When do flat holomorphic connections exist?

When $$E$$ is a locally trivial finite rank sheaf with a connection $$\nabla$$, there is an exact sequence

$$0 \rightarrow \operatorname{End}_{\mathcal{O}_X}(E) \rightarrow \Theta_X(E, \nabla) \rightarrow \Theta_X \rightarrow 0$$

which splits iff $$E$$ has a flat connection. By definition

$$\Theta_X(E, \nabla):= \operatorname{End}_{\mathcal{O}_X}(E)\oplus \Theta_X$$

with the following Lie-structure:

If $$x,y \in \Theta(U)$$ and $$\phi, \psi \in \operatorname{End}_{\mathcal{O}_X}(E)(U)$$, define

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

where $$R_{\nabla}$$ is the curvature of $$\nabla$$.

Note: When you add a potential $$P$$ to a flat connection $$\nabla$$, it does not follow that the new connection $$\nabla+P$$ is flat.

Example: If $$X\subseteq \mathbb{P}^n_{\mathbb{C}}$$ is a smooth quasi projective variety with a finite rank locally trivial sheaf $$E$$ with an algebraic connection $$\nabla$$, you get an extension class

$$na(E,\nabla) \in \operatorname{Ext}^1(\Theta_X, \operatorname{End}_{\mathcal{O}_X}(E))$$

and there is a canonical map

$$c:\operatorname{Ext}^1(\Theta_X, \operatorname{End}_{\mathcal{O}_X}(E)) \rightarrow \operatorname{H}^1(X, \operatorname{End}_{\mathcal{O}_X}(E) \otimes \Omega^1_X),$$

with $$na(E,\nabla)="0"$$ iff $$E$$ has a flat algebraic connection $$\nabla':=\nabla+P$$. The two classes $$c(na(E,\nabla))$$ and $$a(E)$$ live in the same vector space. One may wonder if these two classes are related: If $$c(na(E,\nabla))$$ is a multiple of $$a(E)$$, it follows $$a(E)=0$$ may imply $$na(E,\nabla)=0$$. The "non-abelian cohomology set"

$$\operatorname{Ext}^1(\Theta_X, \operatorname{End}_{\mathcal{O}_X}(E))$$

does not have the structure of an abelian group in general. I believe it has been conjectured that if $$a(E)=0$$ it follows $$na(E,\nabla)=0$$ but do not have a precise reference.

Example: For a line bundle it follows $$\operatorname{End}(L) \cong \mathcal{O}_X$$ and you get an exact sequence

$$NA.\text{ }0 \rightarrow \mathcal{O}_X \rightarrow \Theta_X(L, \nabla) \rightarrow \Theta_X \rightarrow 0.$$

By the above argument it follows this sequence splits over the complex numbers.

Example: Let for simplicity $$L\in \operatorname{Pic}(A)$$ be an invertible $$A$$-module with a connection $$\nabla: T \rightarrow \operatorname{End}_k(L)$$ with $$T:=\operatorname{Der}_k(A)$$. Let $$ad\nabla: T \rightarrow \operatorname{End}_k(L^* \otimes_A L) \cong \operatorname{End}_k(A)$$. It follows for any endomorphism $$\phi_a \in L^*\otimes L$$ we have

$$ad\nabla(x)(\phi_a)=\phi_{x(a)},$$

where $$\phi_a(u):=au$$ for $$u\in L$$. It follows $$ad\nabla$$ is a flat connection. We get an extension

$$NA1.\text{ } 0 \rightarrow L^* \otimes L \rightarrow L^* \otimes L \oplus T \rightarrow^p T \rightarrow 0$$

where the middle term has the following Lie structure:

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

for $$a,b\in L^* \otimes L, x,y \in T$$.

Lemma: A section $$s: T\rightarrow A\oplus T$$ with $$s(x):=(\rho(x),x)$$ is $$A$$-linear and a map of $$k$$-Lie algebras iff the map $$\nabla^*:=\nabla+\rho$$ is a flat connection.

Hence sequence $$A1$$ is the obstruction for the existence of a connection and $$NA1$$ is the obstruction for the existence of a flat connection.

If $$(L,\nabla)$$ is a flat connection on a line bundle $$L$$ it follows sections $$\rho$$ of $$NA$$ are in 1-1 correspondence with flat connections $$\nabla^*:=\nabla+\rho$$. Hence you may view the "set of sections" of $$NA$$ as the "parameter-space of flat connections". There is an algebraic group/a group scheme acting on this parameter space and you may form the "(stack) quotient".

If $$X$$ is a complex projective manifold it follows $$NA$$ should have a section for any $$L\in Pic(X)$$.

Note: Let $$X \subseteq \mathbb{P}^n_S$$ be projective over $$S:=Spec(A)$$ with $$A$$ a finitely generated algebra over a field $$k$$ and let $$E$$ be a coherent $$\mathcal{O}_X$$-module. It follows

$$\operatorname{H}^0(X, \operatorname{End}(E)\otimes \Omega^1_{X/S})$$

is a finitely generated $$A$$-module. This is Hartshorne, Thm II.5.19. In particular if $$A$$ equals $$k$$, it follows the group is a finite dimensional $$k$$-vector space.

Note: The field $$k$$ may be arbitrary, hence the results are valid for a real algebraic variety $$X$$ and a finite rank vector bundle $$E$$ on $$X$$. The "Serre-Swan theorem" gives an equivalence of categories between the category of real smooth finite rank vector bundles on a real smooth manifold $$M$$ and finite rank projective $$R:=C^{\infty}(M)$$-modules. The ring $$R$$ is a commutative unital algebra over the field of real numbers, and the sequences $$A1,NA1$$ exist for any $$R$$-module $$L$$.

• It seems like you're answering about connections in general, while the question was about flat connections specifically. – Gro-Tsen May 26 at 14:20
• @Gro-Tsen - there is a non-abelian Atiyah sequence $NA$ which is split iff the line bundle $L$ has a flat connection. This sequence is not mentioned in the original answer. In fact the accepted answer claims that if $\nabla$ is flat and you add a potential $P$, it follows the new connection $\nabla^*:=\nabla+P$ is flat - this is not correct in general. – hm2020 May 27 at 12:53