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
Special cases and connections: when dim X = 1, HRR specializes to the classical Riemann–Roch for curves,
Historically, Hirzebruch introduced the topological form of the theorem in the 1950s, and later Grothendieck and