## Chern connections in the context of CR-geometry

This post is a result of several discussions with Rodion Déev.

#### 1. Preliminaries: Chern’s theorem

Fix a complex manifold and a complex vector bundle over . Recall that a structure of holomorphic bundle on is given by an operator

which satisfies -Leibniz identity and the integrability conditionFor the reader who is more accustomed to the language of sheaves, the sheaf of holomorphic sections is thus given as . Vice versa if , then is induced by

for any .**Definition**. Let be a Hermitian metric on . A *Chern connection* (with respect to ) is a connection on , such that

- ;
- .

If is a Chern connection, then defines a holomorphic structure on . Chern’s theorem says that the converse is also true, that is:

**Theorem**. (*Chern*). For any holomorphic vector bundle and a Hermitian metric on there exists a unique Chern connection with

Note that if is a Chern connection, its curvature is a -form (its -part vanishes from the definition and part by duality). This observation plays great role in Kähler geometry. For example, if is a very ample bundle in Kähler manifold, its Chern cuvature is a Kähler form representing the corresponding polarisation.

Most of the proofs of Chern’s theorem in standard textbooks consist of more or less ugly coordinate calculus. Now I am going to try and motivate Chern connections geometrically, which will lead us to the proof of the theorem in the case of line bundles and some interesting characterisation of ample line bundles.

#### 2. CR-structures

**Definition**. Let be a smooth manifold. An *almost CR-structure * is simply a codimension subbundle and a complex structure operator . Equivalently, this is a subbundle such that and .

As usual, given such we can put and is the complexification of some real subbundle , on which

acts. Vice versa, the decomposition can be recovered from the eigenspace decomposition for .An almost CR-structure is called* integrable* (or simply a *CR-structure*) if . The letters ”CR” stand either for ”Cauchy-Riemann” or ”Complex-Real.” Whatever interpretation you choose, it is motivated by the following example:

**Example**. Let be a complex manifold and a real hypersurface. Then defines a CR-structure on .

Of course not every CR-structure arises this way. The question whenever a given CR-manifold is embedded (i.e. can be realised as a real hypersurface inside a complex manifold) can be pretty subtle.

One might notice that the condition is in fact unnecessary. We shall say that is a *generalised CR-structure* if the same condition holds, although .

#### 3. Chern connections and Chern-Ehresmann connections

Now let again again be a complex vector bundle of rank over a complex manifold , and a Hermitian metric on it. Denote by the bundle of -unit spheres in . We will usually assume that both and are fixed and use simplified notation . Let be the projection.

*Definition*: A **Chern-Ehresmann connection** on is a generalised CR-structure on , which satisfies the following properties:

- is an isomorphism;
- ;
- the bundle is invariant under the fiberwise -action on .

Note from the first condition that it follows that . In particular, a Chern-Ehresmann connection is a CR-structure iff is a line bundle.

**Lemma**. There exists a bijection between Chern-Ehresmann connections on and -Chern connections on .

Before we prove the lemma we would like to recall the relation between affine connections and equivariant Ehresmann connections; indeed, the lemma is just a complexified (in both senses) version of the same story.

Let be any vector bundle over a smooth manifold . An affine connection is an operator

satisfying the Leibniz identity An Ehresmann connections is a subbundle transversal to .Each affine connection induces the Ehresmann connection, which is a distribution of horizontal subspaces with . Thus splits as with and for each section we get a map

which is inverse to . The curvature of can be computed asVice versa, if is an Ehresmann connection on , which is invariant under the fiberwise action of (here ), one can define the affine connection

*Proof of the Lemma*: The proof is essentially a repeat of the proof of the correspondence between affine connections and Ehresmann -invariant connections. The only specific in here is to observe that the integrability condition is equivalent to .

In details:

Let be a Chern connection on . Its -part defines holomorphic structure on . The total space of the holomorphic vector bundle is a complex manifold itself and the projection is holomorphic.

is embedded as a real hypersurface.

Let be the Ehresmann connection, corresponding to . First of all, since preserves , the horizontal distribution is tangent to the submanifolds , including . Thus, it defines a generalised almost CR-structure on with the complex operator on established by fiberwise isomorphism .

We claim that . Indeed, the space of sections of is generated (over functions) by the sections of the form for some . Observe that from the very construction of complex structure on the map preserves the -decomposition. For any two we have

Since , the summand vanishes. Since the complex structure on is integrable, , and thus .

Finally is -invariant, because is.

In reverse, assume that is a Chern-Ehresman connection. Using the action of we can extend it to the whole (in fact it is well defined only outside the zero section, but on zero section we can just put . Since projects isomoprhically on under , this can be glued into a global smooth distribution on ).

From the singular value decomposition is -invariant, and thus defines an affine connection . By construction it is also -invariant, hence preserves . Finally, similar arguments show that implies that is a Chern connection.

**Corollary.** Chern’s theorem is true for line bundles.

*Proof*: Let be a holomorphic line bundle over and is a metric on . The submanifold is a hypersurface. Consider the distribution on (the complex structure on encodes the holomorphic structure on ). This is a CR-structure on .

Let us check that it is a Chern-Ehresmann connection.

First of all, acts on by holomorphic transformations, therefore preserves the distribution of -vectors on . Therefore, also preserves .

The projection is holomorphic, therefore

Finally, assume that is a vector in the kernel of (here ). Then is a vector tangent to the circle inside the complex line . But for dimension reasons. Therefore and projects to isomorphically.

What is left is to check that the Chern connection corresponding to defines the same holomorphic structure on as . In fact this is a tautology, since a vector field, horizontal with respect to this connection, belongs to iff it is in .

#### 4. Levi forms and Chern’s curvature

Let be a CR-manifold. Consider the real line bundle . If is oriented, then the first Stiefel-Whintey class

vanishes ( because is a complex vector bundle), and therefore we can trivialise it. The map thus can be seen as a -form . This is known as the*Levi form.*

Observe that and is a symmetric bilinear form on . Both Levi forms and the signature of play an important role in complex analysis. Say, if is an open subset inside a complex manifold bounded by a smooth real hypersurface , then is holomorphically convex if and only if the Levi form on is positive.

Again, let be a holomorphic line bundle over and consider with its CR-structure as before. Since it is -invariant, its Levi form is also invariant and descends to a -form on , in fact, this is automatically a -form. From the construction it is clear that this form is precisely the curvature of the underlying Chern connection.

**Corollary**. Let be a holomorphic line bundle over . Choose a hermitian metric on it and put (this is the total space of unit discs fibration over ). Then is anti-ample if and only if is holomorphically convex.