GrothendieckRiemannRoch

The Grothendieck–Riemann–Roch theorem is a fundamental result in algebraic geometry that generalizes the classical Riemann–Roch theorem for curves to higher-dimensional varieties. It was formulated by Alexander Grothendieck in the 1950s as part of his development of sheaf cohomology and algebraic geometry.

In its simplest form, the theorem relates the Euler characteristic of a coherent sheaf on a smooth projective variety to its Chern classes and the Todd class of the tangent bundle. For a smooth projective variety \( X \) over a field and a coherent sheaf \( \mathcal{F} \), the theorem states that the alternating sum of the dimensions of the cohomology groups of \( \mathcal{F} \) is given by the integral of the Chern character of \( \mathcal{F} \) multiplied by the Todd class of the tangent bundle \( TX \) over \( X \). Mathematically, this is expressed as:

\[

\chi(\mathcal{F}) = \int_X \text{ch}(\mathcal{F}) \cdot \text{Td}(TX)

\]

where \( \chi(\mathcal{F}) \) denotes the Euler characteristic of \( \mathcal{F} \), \( \text{ch}(\mathcal{F}) \) is its Chern character, and \( \text{Td}(TX) \) is the Todd class of the tangent bundle. The theorem provides a powerful tool for computing cohomological invariants of sheaves in terms of topological data of the variety.

The Grothendieck–Riemann–Roch theorem plays a crucial role in modern algebraic geometry, particularly in the study of moduli spaces, vector bundles, and intersection theory. It also connects algebraic geometry with topology through the use of characteristic classes and cohomology theories. The theorem was later extended to more general contexts, including derived categories and non-commutative geometry, further solidifying its importance in mathematics.