Hodge

Hodge is a surname of Scottish origin and a term associated with several concepts in mathematics, most notably in differential and complex geometry. The best-known bearer is William Vallance Douglas Hodge (1903–1975), a mathematician whose work laid the foundations of what is now called Hodge theory.

In mathematics, Hodge theory refers to a framework that relates differential forms, cohomology, and metric structures on manifolds. For a smooth, compact manifold with a Riemannian metric, de Rham cohomology classes can be represented by harmonic forms, establishing a bridge between topology and analysis. In the context of complex geometry, Hodge theory yields a decomposition of cohomology into Dolbeault components, giving the Hodge decomposition and leading to Hodge numbers h^{p,q}, which record the dimensions of these components. The Hodge star operator is a metric-dependent map that plays a key role in defining harmonic forms and inner products on forms.

Several important results and conjectures bear Hodge’s name. The Hodge conjecture posits that certain rational cohomology classes on projective varieties arise from algebraic cycles. The Hodge index theorem concerns the signature of the intersection form on divisors of a smooth projective surface. Related concepts include the Hodge filtration and the broader framework of Hodge structures, which describe how complex geometry influences cohomology. Hodge theory has wide applications in algebraic geometry, differential geometry, and mathematical physics. The surname Hodge may also refer to other individuals who bear the name.