Hirzebruch–Riemann–Roch theorem

The Hirzebruch–Riemann–Roch theorem expresses a vector bundle’s holomorphic Euler characteristic on a smooth projective variety as the integral of its Chern character times the variety’s Todd class.

Holomorphic Euler characteristic: The alternating sum of dimensions of a coherent sheaf’s cohomology groups. This is the numerical invariant the theorem computes.
Smooth projective variety: A nonsingular algebraic variety that admits a projective embedding. The theorem relates bundle cohomology and characteristic classes on this space.
Bernhard Riemann: A German mathematician whose work shaped complex analysis, geometry, and number theory. The theorem generalizes the curve-level dimension formula associated with Riemann.
Hirzebruch–Riemann–Roch for curves: The curve case expressing a bundle’s Euler characteristic through its rank, degree, and the curve’s genus. It reduces the general formula to a familiar numerical dimension calculation.
Grothendieck–Riemann–Roch theorem: A theorem relating pushforwards of coherent sheaves to characteristic classes under a proper morphism. It generalizes the setting from a variety’s identity map to a morphism.
Chern character: A characteristic class that converts vector-bundle data into cohomology, compatibly with direct sums and tensor products. It encodes the bundle in the integrand.
Holomorphic vector bundle: A vector bundle whose local trivializations and transition maps are holomorphic. Its holomorphic Euler characteristic is the theorem’s input invariant.
Gustav Roch: A German mathematician known for his collaboration with Bernhard Riemann on divisor theory. The Riemann–Roch theorem underlying the result bears his name.
Geometric genus: The dimension of a smooth variety’s space of global holomorphic top-degree forms. Euler-characteristic formulas can recover genus invariants from characteristic numbers.
Baker–Akhiezer function: A meromorphic function with prescribed poles and specified local behavior, used in integrable systems and algebraic geometry. Unlike this numerical theorem, it constructs functions from spectral-curve data.

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