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The Hodge Decomposition is the following result:

Let $X$ be a smooth projective variety over $\mathbf{C}$. For every integer $k \geq 0$, we have a direct sum decomposition $$H^k(X, \mathbf{C}) \cong \bigoplus_{p+q=k} H^{p,q}(X).$$

My question is: why is this result profound? What "deep underlying truth" about the topology of complex varieties does this theorem reveal?

My guess is that the Hodge decomposition relates the topological structure of $X$ with the complex structure of $X$. The left side of the above direct sum is defined purely topologically (it is singular cohomology). Whereas the right side is defined in terms of the complex structure (it is Dobeault cohomology). So the theorem says that the "complex structure of $X$ sees the entire topological structure of $X$." Is this the right way to view this theorem, or is there another way to see its importance?

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    $\begingroup$ This question may be a bit broad. The last paragraph basically gets at the point of it, however. $\endgroup$
    – yearning4pi
    Commented Dec 25, 2021 at 21:53
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    $\begingroup$ Or, perhaps more interestingly, even though $h^{p,q}$ may vary in a family of diffeomorphic, but holomorphically distinct, manifolds, the sum with $p+q=k$ must stay constant. $\endgroup$
    – Ted Shifrin
    Commented Dec 25, 2021 at 22:14
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    $\begingroup$ One general observation is that this type of geometry-topology theorem theorem deals with various differential operators on bundles, which are in some sense local and depend crucially on the smooth/complex/metric/etc. structures, and shows that we can extract from them topological quantities which are global and depend only on the underlying topological space. The fact that all of this extra information "cancels out" in such a convenient way is at least surprising. $\endgroup$
    – Kajelad
    Commented Dec 26, 2021 at 1:29
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    $\begingroup$ One comment is that it is one of the first phenomena of how local properties on Kahler manifolds pass to cohomology. The "local" version of the Hodge decomposition, that is $\bigwedge^k T_{X,x}^* = \bigoplus_{p+q = k} \bigwedge^{p,q} T_{X,x}^*$, holds on any complex manifold, but something similar only happens on the level of cohomology for Kahler manifolds. Other similar things are for example identities involving Chern forms, like how the integral of the scalar curvature of a Kahler metric equals an intersection number involving the first Chern class. $\endgroup$
    – Gunnar Þór Magnússon
    Commented Jan 5, 2022 at 18:06

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