Hodge theory

Hodge theory is the study of the cohomology of complex and Kähler manifolds through the lens of harmonic differential forms. Introduced by W. V. D. Hodge in the 1930s, it reveals that on a compact Kähler manifold, every cohomology class has a unique harmonic representative — a form that is simultaneously closed and coexact. This leads to the Hodge decomposition, which splits cohomology into components of type (p,q) and imposes severe constraints on the topology of Kähler manifolds. Hodge theory is the bridge between the analysis of partial differential equations on manifolds and the algebraic geometry of their complex structures.