Kähler manifolds - 1

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For a hermitian complex manifold, there is a unique complex connection on the holomorphic tangent bundle compatible with the hermitian metric. Meaanwhile, as a Riemannian manifold, there is a unique torsion-free connection on the real tangent bundle compatible with the induced Riemannian metric. The notion of Kähler manifolds then arises from the comparison of these two connections.

Kähler condition

Suppose is a complex manifold and is a complex connection on the holomorphic tangent bundle . For a local holomorphic coordinates of , we can express locally as We call a torsion-free complex connection if for any local coordinates we have holds for any , and . It can be verified that this is independent of the choice of coordinates by the transformation formula for the connection matrix .

Now specify to the case when admits a hermitian metric . We call the hermitian manifold a Kähler manifold if the associated metric connection is a torsion-free connection on . The metric is then called the Kähler metric on and the associated fundamental form of is called the Kähler form on .

Proposition. Suppose is a complex manifold with a hermitian metric , and is the associated fundamental form of . Then is a Kähler metric if and only if .

Proof: Consider the local coordinates of . Let and the inverse matrix of . By the expression of the metric connection, we obtain Thus is torsion-free if and only if which is equivalent to

Meanwhile, as we see that Therefore if and only if Note that by the hermitian property of , we have The equivalence of the two conditions then follows.

Another important equivalent description of Kähler metrics is that they are almostly the Euclidean metric locally.

Proposition. Suppose is a complex manifold with a hermitian metric . Then is a Kähler metric if and only if for each , there is a coordinate system centered at such that is expressed locally as

Proof: One direction is clear: if has the above local expression, then we see that is Kähler. Conversely, suppose that is a Kähler metric on . Fix a point and consider a local coordinate system centered at with expressed as By the Gram-Schimdt process, we may assume that for each . Then Now define by We can verify that is a well-defined coordinate system locally centered at . Then and hence Note that direct computation yields that

Examples and basic properties

There are several examples of Kähler manifolds whin our familiar complex manifold.

  • The affine space together with the Euclidean metric is Kähler.
  • For a lattice generated by real-linearly independent vectors, the complex torus with the Eucliden metric is Kähler.
  • Suppose is a Riemann surface, i.e., a complex manifold of dimension one. For any hermitian metric with the associated fundamental form , is a -form on , which must be zero. Therefore is a Kähler manifold.
  • If and are both Kähler manifolds, then as the associated fundamental form of the induced metric on is given by we see that is also Kähler.
  • Suppose is a Kähler manifold and is a complex submanifold with the induced metric. Then , where is the embedding, implying that is a Kähler manifold as well.
  • We show that the projective space , together with the Fubini-Study metric , is a Kähler manifold. Let be the real differential operators given by Then and hence locally we have Thus is -closed everywhere.
  • By the previous two results, any compact complex manifold that can be embedded into admits a Kähler metric.

The Kähler condition actually implies some topological properties of .

Proposition. Suppose is a compact Kähler manifold. Then the holomorphic -forms maps injectively into .

Proof: Suppose is a holomorphic -form. We first show that implies that . Let be a local orthonormal frame of with dual frame . Suppose then we have Note that

we obtain the local expression

It follow that whever is nonzero. However, if for some , then Stokes’ formula implies that which can only hold when .

Now we see that is a holomorphic -form. Applying the previous arguments to , we obtain .

Altogether we conclude that there is a natural well-defined injection .

Recall that the Betti number is defined to be the dimensional of the de Rham cohomology:

Propostion. Suppose is a compact Kähler manifold, then for any .

Proof: It suffices to find a closed -form which is not exact for each . Consider , whose closedness follows from the Kähler condition. If for some form , then a contradiction. Thus cannot be exact.