On vector bundles (not neccesarily holomorphic) on a complex manifold, we may consider metrics and connections analogously to the real case. These induce the notion of curvature and Chern class and turn out to be important tools in complex geometry.
Hermitian metric on vector bundles and hermitian manifolds
Suppose is a complex vector bundle on a smooth manifold . Then a hermitian metric on is a hermitian inner product on each fiber depending smoothly on . More precisely, for any open subset and smooth sections of on , the map is a smooth function on . A complex vector bundle together with a hermitian metric is called a hermitian vector bundle.
Using local frame of on an open subset , a hermitian metric on is equivalent to a positive definite hermitian matrix consisting of smooth functions on . Applying the partition of unity theorem, we see that any complex vector bundle on a smooth manifold admits a hermitian metric.
Now suppose is a complex manifold. If the holomorphic tangent bundle of admits a hermitian metric , then we also call a hermitian metric on . A complex manifold together with a hermitian metric is called a hermitian manifold.
For each point , the hermitian metric gives a hermitian form or equivalently
Thus can be identified with a section of the vector bundle . If is a local coordinate of , then the hermitian metric can be expressed as where
Consider the composition with the natural -linear isomorphism given by for each , we obtain a bilinear form This bilinear form can also be constructed by viewing as a subspace and extending the hermitian form on to by zero. Thus the real part of gives a symmetric bilinear form which is further an inner product by the positive definite property of , while the imaginary part of gives an alternating form which determines an element in . Therefore we obatin a Riemannian metric on , called the associated Riemannian metric of , and a real differential -form on , called the fundamental form of . Noting that can also be viewed as an alternating hermitian form on for each , we see that is also a -form, i.e.,
We consider the explicit expression of and in local coordinates. Direct computation yields that and that
Suppose is a holomorphic map such that the holomorphic tangent map is injective for any . Then we may define a hermitian metric on by Viewing a hermitian metric as a section of the tensor bundle, we see that is exactly the image of under the pullback Suppose is the fundamental form of . Then it can be verified that .
Consider a local coordinate of such that form an orthonormal basis locally. Such local coordinate exists by the Gram-Schimdt process. With respect to this basis, we see that The associated volumn form to the Riemannian metric is given by Direct computation then yields that Now suppose is a -dimensional complex submanifold of . Then we have the following formula, which is called the Wirtinger theorem,
We could say more about the fundamental form of a hermitian metric. Indeed, any real -form on determines a hermitian form on . Suppose we have a real -form on expressed locally as Then form a hermitian matrix and hence determines a hermitian form on locally. As is actually independent of coordinates, this hermitian form should be independent of coordinates as well. If the corresponding matrix is positive definite everywhere, then actually defines a hermitian metric on . Such is called a positive -form, and we see a correspondence between hermitian metrics and positive -form on .
We end this part with some examples.
The hermitian metric on given as is called the Euclidean metric. The associated Riemannian matric is exactly the Euclidean metric on .
Suppose is a lattice generated by real-linearly independent vectors. Then there is a standard hermitian metric on the complex torus given by which is also called the Euclidean metric.
There is a hermitian metric on called the Fubini-Study metric constructed as follows. Note that it suffices to give a positive -form on . Consider the real -form on given by If is a lifting of the covering map on an open set , then is a real -form on . For two lifting and on and , respectively, there is a nonvanishing holomorphic function such that on , implying that on the intersection Thus these pullbacks glue together to a real -form on the whole . To see that is positive, noting that acts transitively on and leaves invariant, it suffices to show gives a postive definite hermitian form at one point. Using the local coordinate around , we see that at , is given as which is clearly postive definite.
Connections on vector bundles
In analogy to real case, we need connections on vector bundles to compare different fibers.
Suppose is a complex vector bundle on a complex manifold . Consider the sheaf of smooth sections of the tensor bundle , whose sections are usually called -valued -forms, and the sheaf of smooth sections of the tensor bundle , whose sections are called -valued -forms. Noting that is exactly the sheaf of smooth sections of .
A connection on is a sheaf homomorphism satisfying the Leibniz’ rule for any and on an open subset .
We can express a connection in a local frame. Suppose form a local frame of , i.e., they are sections of on an open subset such that form a basis of for each . Then there are differential -forms on such that We obtain a matrix of -forms on , called the connection matrix of with respect to the local frame . Conversely, given a matrix of -forms, we may define a connection on by
The connection matrix depends on the choice of the local frame. Suppose is another frame, with corresponding connection matrix . The relations between two frames is given by a transformation matrix such that Then we have implying that This is the transformation formula of connection matrices.
Using the connection, we can consider the directional derivatives of sections. Suppose is a connection on a complex vector bundle on . For , a section of on a neighborhood of , and a tangent vector , the directional derivative of at along the direction of is given by where the pairing comes from the natural pairing If is instead a tangent vector field near , then gives a section of near .
Suppose is a smooth curve from to . For each , we can define the tangent vector of the curve at to be
If is a section of on a neighborhood of this curve, then the value of only depends on the value of on . If is a section of on such that , then we call a section of parallel along , and the vector is called the parallel displacement is along .
The choice of connections on a vector bundle is in general not canonical. However, if we consider more structure on the vector bundle and require the connection satisfying some compatible condition, then the choice becomes unique.
First suppose is a holomorphic vector bundle on a complex manifold . Using the decomposition we have Thus we may write a connection as with
Note that since is a holomorphic vector bundle, gives a well-defined sheaf homomorphism Indeed, suppose is a local holomorphic frame of and is an -valued -form on , then we can write as We define to be the section To check this is independent of the choice of the frame, consider another holomorphic frame with Then the coefficients are holomorphic functions on . With respect to this new frame, we have and hence This shows that is well-defined.
In particular, we have a sheaf homomorphism A connection on is called a complex connection, i.e., a connection compatible with the holomorphic structure, if . Using this definiton it is not hard to verify the following proposition.
Proposition. Suppose is a holomorphic vector bundle on a complex manifold and is a connection on . Then is a complex connection if and only if with respect to any local holomorphic frame of , the connection matrix of consists of -forms.
Now suppose is further a hermitian holomorphic vector bundle on with hermitian metric . Then induces a hermitian inner product at each . If is a connection on such that the parallel displacement induced by always preserves the inner product of two holomorpic tangent vectors, then we say is compatible with the hermitian metric . We may also verify that is compatibe with if and only if for each holomorphic sections of , we have where the pairing
and
comes from the pairing
Theorem. Suppose is a holomorphic vector bundle on a complex manifold with a hermitin metric . Then there is a unique complex connection on compatible with , called the metric connection on .
Proof: Suppose is a local holomorphic frame of , and let . Assume that such connection exists, and the connection matrix of with respect to is . Then must be -forms. Consider the inverse matrix of , that is Since is compatible with , we have the equation or equivalently, This has a unique solution given by Thus we have a uniquely determined connection on satisfying all the condition.