HirzebruchRiemannRoch

The Hirzebruch–Riemann–Roch theorem, commonly abbreviated HRR, is a fundamental result in complex geometry and algebraic geometry that relates the analytic data of a vector bundle to topological characteristic classes. It provides a topological formula for the holomorphic Euler characteristic of a vector bundle on a smooth, compact complex manifold or, in the algebraic setting, on a smooth projective variety.

Let X be a compact complex manifold of complex dimension n and E a holomorphic vector bundle on X. The holomorphic Euler characteristic is χ(X, E) = ∑i (−1)^i dim H^i(X, E). The HRR theorem states that χ(X, E) equals the integral over X of the product of the Chern character of E with the Todd class of the tangent bundle TX: χ(X, E) = ∫X ch(E) td(TX). Here ch(E) is the Chern character and td(TX) the Todd class, both expressed in terms of the Chern classes of E and TX.

Special cases and connections: when dim X = 1, HRR specializes to the classical Riemann–Roch for curves, giving χ(X, L) = deg L + 1 − g for a line bundle L on a smooth projective curve of genus g. More generally, HRR applies to coherent sheaves via K-theory and is a precursor to Grothendieck–Riemann–Roch (GRR), which extends the formula to proper morphisms and a broader setting.

Historically, Hirzebruch introduced the topological form of the theorem in the 1950s, and later Grothendieck and others placed it within the broader GRR framework. The HRR theorem links algebraic geometry, topology, and index theory, with wide-ranging applications in geometry and mathematical physics.