Hirzebruch-Riemann-Roch theorem

The Hirzebruch-Riemann-Roch theorem is a fundamental result in geometry that connects the shape and structure of complex spaces (called manifolds) to algebraic data. It provides a way to calculate the number of certain geometric objects (like solutions to equations) on these spaces using characteristic classes—special features capturing curvature and topology. Essentially, it translates complex geometric problems into manageable algebraic formulas, bridging geometry, topology, and algebra. This theorem underpins many advances in mathematical fields, deepening our understanding of the intrinsic properties of complex surfaces and higher-dimensional analogs.