In symplectic geometry, Darboux's theorem says that locally, any symplectic manifold of dimension $2n$ looks like symplectic Euclidean space (that is, there is some set of coordinates $(x\_i, y\_i)$ such that the symplectic form looks like $\sum dx\_i \wedge dy\_i$. Equivalently (I think), there are no local invariants of symplectic manifolds. 


Riemannian geometry has an analogous theorem: that a manifold is Euclidean-flat iff the Riemannian curvature is everywhere 0. 


Is there an analogous theorem for Kahler geometry? That is, is there some tensor $A$ determined by the Kahler form such that if $A$ is globally 0, then there is some set of coordinates $(x\_i y\_i)$ such that the Kahler form looks like $\sum dx\_i \otimes dy\_i$? 


Also, are there any other generalizations of the two theorems above?



Kahler version of Darboux's Theorem
Let $(M,g,J)$ be a Kahler manifold. This means that $J$ is a (integrable) complex structure, so that $(M,J)$ is a complex manifold, $g$ is a smooth metric such that $J$ is $g$-orthogonal, i.e.
$g(JX,JY) = g(X,Y)$
for any smooth vector fields $X$, $Y$ on $M$ and, moreover, the Kahler form is closed.


Ok, now let us assume that given an arbitrary point $p \in M$, there exist local holomorphic coordinates $z\_i$ around $p$ such that


$\omega = \frac{i}{2} \sum\_k dz\_k \wedge d\bar{z}\_k$


with $\omega$ being the Kahler form. By the definition of the Kahler form, we have $g(X,Y) = \omega(X, JY)$, so that in the local coordinates $z\_i$, one has therefore


$g = \frac{1}{2} \sum\_k dz\_k \odot d\bar{z}\_k$


Hence $g$ is locally Euclidean, so that the Riemann curvature tensor vanishes identically. I think the OP is more interested in the converse though.


Conversely, suppose $(M,g,J)$ is a Kahler manifold with Riemann curvature vanishing identically. Being flat, given an arbitrary point $p \in M$, the local holonomy group $Hol\_0(g,p)$ is trivial. Thus starting with a basis $\alpha\_1,\cdots,\alpha\_m$ of the space of $1$-forms at $p$ of type $(1,0)$ (with $m$ being the complex dimension of $M$), one can locally extend the $\alpha\_i$ using parallel transport locally to some parallel $\alpha\_i$ defined in some neighborhood of $p$. In particular, these $\alpha\_i$ form a local holomorphic coframe, with each $\alpha\_i$ being further closed. Then using Dolbeault cohomology, it follows that there exist local holomorphic coordinates $z\_i$ such that
$dz\_i = \alpha\_i$.


Moreover, if at $p$, the $\alpha\_i$ were chosen to be unitary (at $p$), it follows that their extensions to a neighborhood are also unitary (and as we have seen, parallel too) so that the metric $g$ can be written locally as:


$ g = \frac{1}{2} \sum\_k \alpha\_k \odot \bar{\alpha}\_k = \frac{1}{2} \sum\_k dz\_k \odot d\bar{z}\_k$


Thus the tensor you are looking for is the Riemann curvature tensor. 


