Hodge theory relates the topology of a smooth manifold to differential equations on differential forms. On a compact oriented
Riemannian manifold, every real cohomology
class has a unique harmonic form representative.
On a compact Kähler manifold, complex cohomology further decomposes into Dolbeault groups of
types 
.
Hodge Theory
See also
Dolbeault Cohomology, Harmonic Form, Hodge Diamond, Hodge StarExplore with Wolfram|Alpha
References
Wells, R. O. Jr. Differential Analysis on Complex Manifolds, 3rd ed. New York: Springer-Verlag, 2008.Cite this as:
Weisstein, Eric W. "Hodge Theory." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HodgeTheory.html