Ytukyg

I am really struggling to understand the basics of Kahler geometry and hope someone can give me some guidance. Suppose we have a complex manifold with some complex structure $J$ and let $g$ be a Hermitian metric. That is, $g$ is a Riemannian metric and satisfies the extra condition $g(JX,JY)=g(X,Y)$. So we have a Riemannian manifold $(M^{2n},g)$. Why do we all of a sudden start considering the complexified tangent space? What good does that do? The metric $g$ is of the form $$g=sum_{i,j=1}^{2n}g_{ij}E^iotimes E^j$$ where ${E^1,dots, E^{2n}}={dx^1,dy^1,dots, dx^n, dy^n}$. We introduce $frac{partial}{partial z_k}$ and $frac{partial}{partial overline z_k}$ and $dz_k$ and $doverline z_k$ and then extend the metric to $TM^mathbb{C}$ and start doing all these computations. But by doing this we've completely changed the domain of the metric. One metric is in $Omega^2_mathbb{C}(M)$ and the other is in $Omega^2(M)$. How is this new metric telling us stuff about the original one? What is the relationship between the matrix $g_{koverline j}$, where $1leq j,kleq n$ and $g_{jk}$, where $1leq j,kleq 2n$? We introduce the 'extended' connection and curvature but again, I don't see how this new information tells us something about the original manifold. For example, apparently the 'extended' definition of Ricci curvature on $TM^mathbb{C}$ is the same as the Ricci curvature on $(M^{2n},g)$. How can these two things be the same, they are functions on completely different spaces?? Another example I am struggling with is how the Kahler form $omega$ satisfies $frac{omega^n}{n!}=text{vol}_M$. We have $omega=sqrt{-1}g_{joverline k}dz^jwedge doverline z^k$ an element of $Omega^2_mathbb{C}(M)$. How can this be raised to powers to give an element of
$Omega^{2n}(M)$??

Really any help would be greatly appreciated!

Kähler manifolds are particularly nice examples of manifolds where the complex structure and the Riemannian structure coalesce perfectly. Every complex manifold may be viewed as a Riemannian manifold (since holomorphic charts are automatically smooth), but it is not necessarily the case that these two structures will be compatible. This compatibility can be expressed in a number of ways:

• We have a notion of parallel transport on Riemannian manifolds. A certain compatibility criterion between the complex and Riemannian structures would be to require that parallel transport commutes with $J$.




• On the tangent bundle of a complex manifold, we have two choices of canonical connection, the Levi--Civita connection from Riemannian geometry and the Chern connection from complex geometry. These two connections do not necessarily coincide on an arbitrary complex manifold.




• On a Riemannian manifold we may diagonalise the metric at a point which gives us the Riemannian Normal Coordinates. On an a complex manifold we have these coordinates also (of course), but these coordinates are not necessarily holomorphic.

PUNCHLINE: On a Kähler manifold all of these conditions are satisfied! In fact, the Kähler condition $domega =0$ is equivalent to the three conditions above (any of which may be taken as a definition).

Closing remarks: It is worth noting that the condition $g(JX, JY) = g(X,Y)$ may be stated as requiring the complex structure $J$ to be an orthogonal transformation on each tangent space.

Commuting with the complex structure $J$ is desirable for the same reason that holomorphic functions are desirable relative to smooth functions. That is, a function $f : mathbb{C} longrightarrow mathbb{C}$ is holomorphic if its Jacobian matrix commutes with $J = begin{bmatrix}
0 & -1 \
1 & 0
end{bmatrix}$.

Notice that this also alludes to why holomorphic functions are such starkly different beasts when compared with smooth functions. This requirement on the derivative is quite rigid.

References: Griffiths and Harris, Principles of Algebraic Geometry, Andrei Moroianu Lecture on Kähler Geometry, Werner Ballmann Lectures on Kähler manifolds, Gabór Szekelyhidi Introduction to Extremal Metrics.

This is also nice: Key differences between almost complex manifolds and complex manifolds

Updated: I have recently been made aware of Daniele Angella's text Cohomological Aspects in complex Non-Kähler geometry. As the name suggests, the book focuses on complex manifolds which are not Kähler, but I would highly recommend this well-written text. It has a nice treatment of almost complex manifolds, and it is often insightful to look at the behaviour of the objects which do not satisfy such a nice condition as being Kähler.

Customary we write Kahler metric $$ ds^2=2sum_{a,b}
g_{aoverline{b}}dz_aotimes doverline{z}_b
$$

Here $g$ is a Riemannian metric. If $E_a=frac{partial }{partial
x_a}, E_{n+a}=frac{partial }{partial x_{n+a}}$ are coordinate
field, then we have dual $dE_a, dE_{n+a}$.

In further, define $$ frac{partial
}{partial z_a} =frac{partial }{partial x_a} -ifrac{partial
}{partial x_{n+a}} $$ so that its dual is $dz_a=frac{1}{2}{ dE_a+
idE_{n+a} }$.

If $g_{aoverline{b}} :=g(frac{partial }{partial
z_a},overline{frac{partial }{partial z_b}} )$, then by a direct
computation considering $J$, we have $$ g=2{rm Re} sum_{a,b}
g_{aoverline{b}}dz_aotimes doverline{z}_b = {rm Re} ds^2 $$

In further, fundamental form is $Phi (X,Y)=g(X,JY)$. By
computation, we have $Phi = {rm Im} ds^2 $.