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Chern connections in the context of CR-geometry

1.Preliminaries: Chern theorem

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

Fix a complex manifold X and a complex vector bundle E over X.

Recall that a structure of holomorphic bundle on E is given by an operator

    \[\overline{\partial}_{\mathcal{E}} \colon \Gamma (E) \to \Gamma(E \otimes \Lambda^{0,1}X)\]

which satisfies \overline{\partial}-Leibniz identity

    \[\overline{\partial}_{\mathcal{E}}(fs) = f\overline{\partial}_{\mathcal{E}}(s) + \overline{\partial}(f) \otimes s\]

and the integrability condition

    \[\overline{\partial}_{\mathcal{E}}^2 = 0.\]

For the reader who is more accustomed to the language of sheaves the sheaf of holomorphic sections is thus given as \mathcal{E} := \operatorname{Ker}(\overline{\partial}_{\mathcal{E}}). Vice versa if E = \mathcal{E} \otimes_{\mathcal{O}_X} \mathcal{C}^{\infty}(X), then \overline{\partial}_{\mathcal{E}} is induced by

    \[\overline{\partial}_{\mathcal{E}}(f \otimes s) = \dibar f \otimes s\]

for any s \in H^0(X, \mathcal{E}).

Definition. Let h be a Hermitian metric on E. A Chern connection (with respect to h) is a connection \nabla on E, such that

  1. \nabla h = 0;
  2. (\nabla^{0,1})^2 = 0.

If \nabla is a Chern connection, then \nabla^{0,1} defines a holomorphic structure on E. The Chern theorem says that the converse is also true, that is

Theorem (Chern). For any holomorphic vector bundle (E, \overline{\partial}_{\mathcal{E}}) and a Hermitian metric h on E there exists a unique  Chern connection \nabla with \nabla^{0,1} = \overline{\partial}_{\mathcal{E}}

Note that if \nabla is a Chern connection, it’s curvature is a (1,1)-form (it’s (0,2)-part vanishes from the definition and (0,2) part by duality). This observation plays great rôle in Kähler geometry. For example, if \mathcal{E} is a very ample bundle in Kähler manifold, it’s Chern cuvature is a Kähler form represents the corresponding polarisation.

Most of the proofs  of Chern theorem with which you can come up in textbooks consist of more or less ugly coordinate calculus. Now I am going to bring some geometric picture behind Chern connections which will lead us to the proof of the theorem in case of line bundles and some interesting characterisation of ample line bundles.


Definition. Let M be a smooth manifold. An almost CR-structure (H, J) is simply a codimension 1 subbundle H \subset TM and a complex structure operator J \colon H \to H, J^2 = -1. Equivalently, this is a subbundle H^{0,1} \subset TM \otimes \mathbb{C} such that H^{0,1} \cap \overline{H^{0,1}} = 0 and \dim H^{0,1} = \frac{\dim M -1}{2}.


As usual, given such H^{0,1} we can put H^{1,0} := \overline{H^{0,1}} and H_{\mathbb{C}} = H^{1,0} \oplus H^{0,1} is the complexification of some real subbundle H \subset TM, on which

    \[J = \sqrt{-1}\operatorname{Id}_{H^{1,0}} - \sqrt{-1}\operatorname{Id}_{H^{0, 1}}\]

acts. Vice versa, the decomposition H \otimes \mathbb{C} = H^{1,0} \oplus H^{0,1} can be recovered as eigenspace decomposition for J.

An almost CR-structure is called integrable (or simply a CR-structure) if [H^{0,1}, H^{0,1}] \subset H^{0,1}. The letters ”CR” stand either for ”Cauchey-Riemann” or ”Complex – Real ”. Whatever interpretation you choose, it is motivated by the following example:

Example. Let Z be a complex manifold and M \subset Z a real hypersurface. Then H^{0,1} := T^{0,1}Z \cap TM defines a CR-structure on M.

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 \codim H = 1 is in fact unnecessary. We shall say that H is a generalised CR-structure if the same condition holds, although \codim H > 1.


3.Chern connections and Chern-Ehresmann connections

Now let again E be again a complex vector bundle of rank n over a complex manifold X and h a Hermitian metric on it. Denote by \Sigma_h(E) the bundle of h-unit spheres in E. We will usually assume that f both E and h are fixed and use simplified notation \Sigma: = \Sigma_h(E). Let p \colon \Sigma \to X be the projection.


Definition: A Chern-Ehresmann connection on E is a generalised CR-structure (H, J) on \Sigma, which satisfies the following properties:

  1. Dp \colon H \to TX is an isomorphism;
  2. Dp(H^{0,1}) \subseteq T^{0,1}X;
  3. the bundle H is  invariant under the fiberwise \operatorname{U}(n)-action on \Sigma

Note that from the first condition it follows that \codim_{\mathbb{R}} H = \rk_{\mathbb{R}}E - 1. In particular, a Chern-Ehresmann connection is a CR-structure iff E is a line bundle.

Lemma. There exists a bijection between Chern-Ehresmann connections on \Sigma_h(E) and h-Chern connections on E.

Before we proof 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 p \colon E \to X be any vector bundle over a smooth manifold X. An affine connection is an operator

    \[\nabla \colon \Gamma(E) \to \Gamma(E \otimes T^*X)\]

satisfying Leibniz identity

    \[\nabla(f s) = f\nabla(s) + df \otimes s.\]

An Ehresmann connections is a subbundle H \subset T\operatorname{Tot}(E) transversal to \operatorname{Ker}(Dp).

Each affine connection \nabla induces the Ehresmann connection, which is a distribution of horizontal subspaces H \subset T\operatorname{Tot}(\mathcal{E}) with Dp \colon H \xrightarrow{\sim} TX.  Thus T\operatorname{Tot}(E) splits as V \oplus H with V = \operatorname{Ker} Dp and for each section s \colon X \to E we get a map

    \[L_s \colon TX \xrightarrow{Ds} T\operatorname{Tot}(\mathcal{E}) = V \oplus H \to H,\]

which is inverse to Dp. The curvature of \nabla can be computed as

    \[\Theta_{\nabla}(\xi_1, \xi_2)(s) = [L_s(\xi_1), L_s(\xi_2)]-L_s([\xi_1, \xi_2])\]


Vice versa, if H is an Ehresmann connection on E, which is invariant under the fiberwise action of \operatorname{GL}_r(\mathbb{R}) (here r = \rk E), one can define the affine connection

    \[\nabla_{\xi}(s) := Ds(\xi) \mod H \in T\operatorname{Tot}(E)/H \simeq \operatorname{Ker} Dp \simeq E\]

Proof of the Lemma: Essentially repeats the proof of the  correspondence between affine connections and Ehresmann \operatorname{GL}_r-invariant connections. The only specific in here is to observe that the integrability condition [H^{0,1}, H^{0,1}] \subseteq H^{0,1} is equivalent to (\nabla^{0,1})^2 = 0.

In details:

Let \nabla be a Chern connection on (E, h). It’s (0,1)-part defines holomorphic structure on E. The total space of the holomorphic vector bundle \operatorname{Tot}(\mathcal{E}) is a complex manifold itself and the projection p \colon \operatorname{Tot}(E) \to X is holomorphic.

\Sigma \subset \operatorname{Tot}(\mathcal{E}) is embedded as a real hypersurface.

Let H be the Ehresmann connection, corresponding to \nabla. First of all, since \nablda preserves h, the horizontal distribution is tangent to the submanifolds \{h(v,v) = c\} \subset \operatorname{Tot}(\mathcal{E}), including \Sigma. Thus, it defines a generalised almost CR-structure on \Sigma with the complex operator on H established by fiberwise isomorphism H \xrightarrow{\sim} TX.

We claim that [\widetilde{H}^{0,1}, \widetilde{H}^{0,1}] \subset \widetilde{H}^{0,1}. Indeed, the space of sections of H is generated (over functions) by the sections of the form L_s(\xi) for some \xi \in \Gamma(TX). Observe that from the very construction of complex structure on H the map L_s preserves the (p,q)-decomposition. For any two \xi_1, \xi_2 \in \Gamma(T^{0,1}X) we have

    \[[L_s(\xi_1), L_s(\xi_2)] = \Theta_{\nabla}(\xi_1,\xi_2) - L_s([\xi_1, \xi_2]\]

Since (\nabla^{0,1})^2 = 0, the summand \Theta_{\nabla}(\xi_1, \xi_2) vanishes. Since the complex structure on X is integrable, [\xi_1, \xi_2] \in T^{0,1}X, and thus L_s([\xi_1, \xi_2]) \in H^{0,1}.

Finally H is \operatorname{U}(n)-invariant, because \nabla is.

In reverse, assume that H is a Chern-Ehresman connection. Using the action of \mathbb{C}^{\times} we can extend it to the whole \operatorname{Tot}(\mathcal{E}) (in fact it is well defined only outside the zero section, but on zero section we can just put H = TX. Since H projects isomoprhically on TX under Dp, this can be glued into a global smooth distribution  \widetilde{H} on \operatorname{Tot}(\mathcal{E})).

From the singular value decomposition \widetilde{H} is \operatorname{GL}_{r}(\mathbb{C}-invariant, and thus defines an affine connection \nabla. By construction it is also \operatorname{U}(r)-invariant, hence preserves h. Finally, similar arguments show that [H^{0,1}, H^{0,1}] \subseteq H^{0,1} implies that \nabla is a Chern connection.



Corollary: Chern theorem is true for line bundles.

Proof: Let \mathcal{E} be a holomorphic line bundle over X and h is a metric on E = \mathcal{E} \otimes_{\mathcal{O}_X} \mathcal{C}^{\infty}(X). The submanifold \Sigma := \Sigma_h(E) \subset \operatorname{Tot}(\mathcal{E}) is a hypersurface. Consider the distribution H^{0,1}:= T^{0,1}\operatorname{Tot}(\mathcal{E}) \cap T\Sigma on \Sigma (the complex structure on \operatorname{Tot}(\mathcal{E}) encodes the holomorphic structure on E). This is a CR-structure on \Sigma.

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

First of all,  \mathbb{C}^{\times} acts on \operatorname{Tot}(\mathcal{E}) by holomorphic transformations, therefore preserves the distribution of (0,1)-vectors on \operatorname{Tot}(\mathcal{E}). Therefore, \operatorname{U}(1) \subset \mathbb{C}^{\times} also preserves H^{0,1} = T\Sigma \cap T^{0,1}\operatorname{Tot}(\mathcal{E}).

The projection p \colon \operatorname{Tot}(\mathcal{E}) \to X is holomorphic, therefore

    \[Dp(H^{0,1}) \subseteq Dp(T^{0,1}\operatorname{Tot}(\mathcal{E})) \subseteq T^{0,1}X.\]

Finally, assume that v \in H_{(x, z)}^{0,1} is a vector in the kernel of Dp (here (x, z) \in \Sigma, \  p((x,z)) = x \in X). Then v is a vector tangent to the circle \Sigma_x := E_x \cap \Sigma inside the complex line E_x. But T^{0,1}E_x \cap T\Sigma_x = 0 for dimension reasons. Therefore H^{0,1} \cap \operatorname{Ker}Dp = 0 and H projects to TX isomorphically.

What is left is to check that the Chern connection \nabla corresponding to H defines the same holomorphic structure on E as \overline{\partial}_{\mathcal{E}}. In fact this is a tautology, since a vector field, horizontal with respect to this connection, belongs to H^{0,1} iff it is in T^{0,1}\operatorname{Tot}(\mathcal{E}).



4. Levi forms and Chern’s curvature

Let (M, H^{0,1}) be a CR-manifold.  Consider the real line bundle TM/H. If M is oriented, then the first Stiefel-Whintey class

    \[w_1(TM/H) = w_1(TM) + w_1(H)\]

vanishes (w_1(H) = 0 because H is a complex vector bundle), and therefore we can trivialise it. The map

    \[\Lambda^2 H \to TM/H, \ \xi_1 \wedge \xi_2 \mapsto [\xi_1, \xi_2] \mod H\]

thus can be seen as a 2-form \omega \in \Gamma(\Lambda^2H^*). This is known as Levi form.

Observe that \omega^{0,2} =0 and \omega(J \cdot, \cdot) is a symmetric bilinear form on H . Both Levi forms and the signature of \omega(J\cdot, \cdot) play important  rôle in complex analysis. Say, if U \subset Z is an open subset inside a complex manifold bounded by a smooth real hypersurface M, then U is holomorphically convex iff the Levi form on M is positive.

Again, let \mathcal{E} be a holomorphic line bundle over X and consider \Sigma \subset \operatorname{Tot}(\mathcal{E}) with its CR-structure  H as before. Since is \operatorname{U}(1)-invariant, it’s Levi form is also invariant and descends to a 2-form on X, in fact, this is automatically a (1,1)-form. From the construction it’s clear that this form is precisely the curvature of the underlying Chern connection.

Corollary. Let \mathcal{E} be a holomorphic line bundle over X. Choose a hermitian metric h on it and put E_{<1}:=\{h(z,z) < 1\} \subset \operatorname{Tot}(\mathcal{E}) (this is the total space of unit discs fibration over X). Then \mathcal{E} is ample if and only if E_{<1} is holomorphically convex.

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