Riemann–Roch theorem for surfaces

A formula for a divisor on a smooth projective surface: its Euler characteristic equals its self-intersection and canonical intersection terms, plus the surface’s Euler characteristic.

Euler characteristic of a coherent sheaf: The alternating sum of the dimensions of a sheaf’s cohomology groups. The theorem computes this invariant for the line bundle associated with a divisor.
Linear system (algebraic geometry): A projective family of divisors linearly equivalent to a fixed divisor. Riemann–Roch estimates the sections that determine the size of this family.
Bernhard Riemann: A German mathematician whose work founded major parts of complex analysis, geometry, and number theory. The theorem’s name traces to Riemann’s original result for curves.
Smooth projective surface: A smooth, two-dimensional algebraic variety embedded in projective space. The theorem assumes this setting so divisor intersections and coherent cohomology behave as stated.
Intersection form: A bilinear pairing that assigns intersection numbers to divisor classes on a surface. The divisor’s self-intersection and canonical intersection supply the formula’s quadratic and linear terms.
Geometric genus: The dimension of the space of global holomorphic two-forms on a smooth projective surface. Surface Riemann–Roch relates this invariant to the surface’s Euler characteristic and canonical class.
Gustav Roch: A German mathematician known for his work on the theorem relating divisors and meromorphic functions on curves. Roch’s contribution completed the curve theorem later generalized to surfaces.
Divisor (algebraic geometry): A formal integer combination of codimension-one subvarieties on an algebraic variety. The theorem takes a divisor as input and associates it with a line bundle.
Canonical divisor: A divisor class representing the line bundle of top-degree differential forms on a smooth variety. Its intersection with the chosen divisor determines the formula’s linear correction.
Kodaira dimension: An invariant measuring the asymptotic growth of pluricanonical sections of a variety. Applying the formula to multiples of the canonical divisor helps detect that growth.

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Riemann–Roch theoremBroader topic: It extends the curve theorem’s dimension-counting principle to surfaces.