Motivation for Kahler Geometry

Question

I have been studying Symplectic Geometry. Previously I studied Riemannian Geometry.

In Symplectic Geometry I learned the existence of an almost complex structure and how some special almost complex manifolds are integrable thereby becoming a complex manifold.

Now I am seeing that we want to focus on a manifold which has all 3 of these structures (symplectic,riemannian and complex) and in some sense are compatible.

My question is : All we did was define special types of 2 forms on a manifold and then started looking at compatibility between them. So there should be no reason to give these forms extra privilege, I could just as well define some weird structure like maybe a closed k-form which satisfies blah-blah properties and that would lead to a new geometry etc.

How does a Mathematician know which structures are "important" and worth our attention?

Answer 1

As you allude to, a Kähler structure indeed consists of three mutually compatible structures:

1. A Riemannian metric $g$, i.e., a smoothly varying positive-definite quadratic form on each tangent space;
2. A symplectic form $\omega$, i.e., a smoothly varying non-degenerate skew-symmetric $2$-form on each tangent space which is closed;
3. A integrable complex structure, i.e., a smoothly varying endomorphism on each tangent which squares to $-\text{id}$ and is integrable.

Any smooth manifold carries an object of type 1. The non-degeneracy assumption in 2, forces the dimension of the underlying manifold to be even. Moreover, the fact that the symplectic form is closed, further imposes a cohomological constraint (that is, the second de Rham cohomology group is necessarily non-trivial). Any manifold which carries structure 2 carries structure 3, albeit, the complex structure may fail to be integrable, i.e., a symplectic manifold carries an almost complex structure. The obstructions to the existence of an almost complex structure are cohomological (specifically, almost complex structures force certain relations on characteristic classes, which can be used to show that $\mathbb{S}^2$ and $\mathbb{S}^6$ are the only spheres which carry almost complex structures). The integrability of these complex structures, however, is more difficult to obstruct, and much less is known. In fact, typically one shows that a complex structure is obstructed by showing that there is a compatible symplectic form, and showing that the existence of the symplectic form is obstructed (e.g., the second de Rham cohomology group vanishes).

Now, what does it mean for such structures to be compatible? Well, we say that a complex structure $J$ is $\omega$--tame, for a given symplectic form $\omega$, if $$\omega(u,Ju) > 0$$ for all non-zero tangent vectors $u$. We say that $J$ is $\omega$--compatible if it is $\omega$--tame, and $$\omega(Ju, Jv) = \omega(u,v)$$ for all tangent vectors $u,v$.

A Riemannian metric $g$ is compatible with a complex structure $J$ if $g$ is $J$--Hermitian, i.e., $$g(Ju, Jv) = g(u,v)$$ for all tangent vectors $u,v$.

Finally, we say that the triple $(g, J, \omega)$ is compatible if $J$ is compatible with $g$ and $\omega$ in the above sense, and $$\omega(u,v) = g(Ju, v).$$

From this point of view, the question is not, these are the right structures to study, but rather, why should such objects exist at all? The marvellous fact, however, is that such structures appear in abundance: $\mathbb{P}^n$ is Kähler, and since the Kähler property is preserved under restriction to complex sub-manifolds, projective manifolds are Kähler; Stein manifolds are Kähler, and there are many more examples.

This is not how Kähler manifolds were introduced, however. The original definition given Erich Kähler was that a complex manifold is Kähler if it carries a Hermitian metric which is locally given by $\sqrt{-1} \partial \overline{\partial} \varphi$ for a pluri-subharmonic function $\varphi$. This seemingly innocent class of Hermitian manifolds turned out to have the rich beautiful properties listed above.

Answer 2

Perhaps, you could remember that Kähler geometry was invented before symplectic geometry. It is "natural", in the sense that it is the natural geometry of complex projective manifolds. As a submanifold of $\bf CP^N$ a complex algebraic manifold is endowed with a Kähler structure, and this structure is the main tool to prove basic results : Hodge Theorems, Hard Lefschetz Theorem , for instance.