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
For 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
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
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
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
Vice 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
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.