Riemann Roch TheoremEdit
The Riemann–Roch theorem sits at a junction of several foundational strands in mathematics: complex analysis, algebraic geometry, and topology. It gives a precise count of how many independent meromorphic functions or sections of a line bundle exist on a compact Riemann surface, once one fixes where those functions are allowed to have poles. In essence, it translates a problem about analytic functions into a statement about global geometric data, and from this translation flow many further insights about the geometry of curves, their embeddings into projective space, and the structure of linear systems.
The theorem comes into its fullest power when one passes from analytic surfaces to the algebraic language of curves over a field. There, the same principle—the balance between poles and holomorphic data—persists, but it is expressed in terms of divisors, line bundles, and cohomology. The Riemann–Roch framework explains why certain divisors on a curve have more global sections than one might naively expect, and it clarifies how the geometry of the curve constrains those sections. The result is a cornerstone of the study of algebraic curves and their maps to projective spaces, and it underpins many constructions in modern algebraic geometry.
Statement
Let X be a compact Riemann surface of genus g, and let D be a divisor on X. Denote by l(D) the dimension of the space L(D) of meromorphic functions on X whose poles are constrained by D; equivalently, l(D) is the dimension of the space of global sections of the line bundle associated to D, i.e., l(D) = dim H^0(X, O(D)). Let K be a canonical divisor on X, which corresponds to the line bundle of holomorphic differentials (the “canonical bundle”). The Riemann–Roch theorem states:
l(D) − l(K − D) = deg(D) + 1 − g.
Here deg(D) is the degree of the divisor D, and g is the genus of X. In the language of cohomology, this is often written as:
h^0(D) − h^0(K − D) = deg(D) + 1 − g.
Certain corollaries follow immediately. For divisors D with deg(D) > 2g − 2, the term l(K − D) vanishes, so l(D) = deg(D) + 1 − g. In the simplest case, when X has genus g = 0 (the Riemann sphere), one recovers the familiar fact that for a divisor D of degree d ≥ 0, l(D) = d + 1, and l(D) = 0 if d < 0.
The theorem can also be stated in a purely algebraic form for curves over arbitrary fields, with the same essential relation between the dimension of global sections of a line bundle and the geometry encoded in D and K. In modern language, it is typically presented through the cohomology of line bundles on X and is closely tied to duality theories such as Serre duality.
History and context
The analytic version of the idea grew from the study of meromorphic functions on compact Riemann surfaces in the 19th century, chiefly under the work of Bernhard Riemann. The original intuition connected the possible poles of a function to the topology (via the genus) of the underlying surface. The algebraic formulation—that the same balance persists for algebraic curves over fields—was developed by subsequent generations of geometers, with significant contributions from Max Noether and others who recast the theory in the language of divisors, line bundles, and cohomology.
In the 20th century, the machinery of sheaf cohomology and duality clarified the foundations. The theorem is now presented in multiple equivalent forms, and it serves as a bridge to powerful generalizations. In particular, the Hirzebruch–Riemann–Roch theorem extends the basic idea to higher-dimensional varieties, and its Grothendieck–Riemann–Roch refinement organizes the interaction of pushforwards of sheaves with characteristic classes in families of varieties.
Variants and related concepts
- Divisors and linear systems: A divisor D encodes prescribed poles and zeros for functions or sections. The set of global sections H^0(X, O(D)) forms a vector space whose dimension is l(D). The collection of effective divisors linearly equivalent to D forms a complete linear system, denoted |D|.
- Canonical divisor and differentials: The canonical divisor K corresponds to holomorphic differentials on X. The degree of K equals 2g − 2, and K plays a central role in the duality appearing in the theorem.
- Line bundles and cohomology: The statement is naturally phrased in terms of line bundles L with L ≅ O(D) and their global sections H^0(X, L). The complementary piece l(K − D) is a cohomological dual, reflecting Serre duality on the curve.
- Special divisors and embeddings: The theorem explains when a divisor D has more sections than expected, leading to the study of special divisors and to embeddings of X into projective space via complete linear systems |D|.
Examples and applications
- Genus 0: If X is the Riemann sphere (g = 0) and D has degree d ≥ 0, then l(D) = d + 1, which matches the classical fact that rational functions with prescribed poles at up to d points form a (d + 1)-dimensional space.
- High-degree divisors: For a curve of genus g and a divisor D with deg(D) > 2g − 2, the Riemann–Roch formula gives l(D) = deg(D) + 1 − g, since l(K − D) = 0 in this range.
- Canonical embedding and syzygies: The Riemann–Roch framework underpins the construction of the canonical map from a non-hyperelliptic curve into projective space, using the global sections of the canonical bundle canonical divisor.
- Weierstrass gaps and gaps sequences: The dimensions l(D) for various D illuminate the gap structure at points of X, leading to the study of special points such as Weierstrass points.
- Generalizations and influence: The theorem informs many techniques in algebraic geometry, including the use of linear systems to study maps to projective spaces and the interplay between geometry and analysis on curves. It also serves as a stepping stone to broader cohomological and categorical frameworks, such as the modern articulation of the Hirzebruch–Riemann–Roch theorem and the Grothendieck–Riemann–Roch theorem.