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 \(X\) 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 \(E\). Chern’s 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, its curvature is a \((1,1)\)-form (its \((0,2)\)-part vanishes from the definition and \((0,2)\) part by duality). This observation plays great role in Kähler geometry. For example, if \(\mathcal{E}\) 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 \(M\) 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 from the 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 ”Cauchy-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 \(E\) again be 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 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 from the first condition that 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 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 \(p \colon E \to X\) 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 the 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: The proof is essentially a repeat of 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 \((E, h)\). Its \((0,1)\)-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. \(\qed\)

Corollary. Chern’s 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 \(T^{0,1}\operatorname{Tot}(\mathcal{E})\). \(\qed\)

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 the 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 an important role 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 if and only if 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 it is \(\operatorname{U}(1)\)-invariant, its 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 is 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 anti-ample if and only if \(E_{<1}\) is holomorphically convex.

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