Cohomology on Algebraic Varieties

Introduction

Cohomology on algebraic varieties is a fundamental tool in modern algebraic geometry that allows mathematicians to study global properties of varieties by examining local behavior. This concept bridges algebraic geometry with topology and differential geometry, providing powerful invariants that characterize geometric objects.

The study of cohomology on algebraic varieties has been one of the most significant developments in 20th-century mathematics, with contributions from luminaries such as Alexander Grothendieck, Jean-Pierre Serre, and Oscar Zariski. These theories have enabled profound insights into the structure of algebraic varieties and have led to solutions of long-standing problems in number theory and geometry.

Foundations of Cohomology

To understand cohomology on algebraic varieties, we first need to establish the basic definitions and constructions. An algebraic variety is a fundamental object of study in algebraic geometry, which can be thought of as the solution set to polynomial equations. Cohomology groups are algebraic invariants that capture information about the global structure of such varieties.

For an algebraic variety X over a field k, we can define various cohomology theories, including:

• Singular cohomology: When X is viewed as a topological space, we can consider its singular cohomology groups Hⁱ(X, ).
• Sheaf cohomology: More intrinsic to algebraic geometry, this associates to a sheaf F on X cohomology groups Hⁱ(X, F).
• tale cohomology: A powerful construction that allows topological intuition to be applied to varieties over arbitrary fields.
• De Rham cohomology: For smooth varieties in characteristic zero, this captures differential forms information.

Sheaf Cohomology

Sheaf cohomology, developed by Jean-Pierre Serre, is particularly important in algebraic geometry. Given a sheaf F of abelian groups on a topological space X, the sheaf cohomology groups Hⁱ(X, F) measure the obstruction to solving global problems using local information.

One fundamental tool in computing sheaf cohomology is the ech cohomology. Given an open covering {U_i} of X and a sheaf F, one can construct the ech cohomology groups ⁱ({U_i}, F). Under suitable conditions on the covering, these ech cohomology groups agree with the sheaf cohomology groups Hⁱ(X, F).

Cohomology of Coherent Sheaves

For coherent sheaves on Noetherian schemes, cohomology groups are finite-dimensional vector spaces when the base field is algebraically closed. This finiteness property is crucial for many applications.

The Hirzebruch-Riemann-Roch theorem relates the analytical and topological invariants of a vector bundle on a compact complex manifold. For a projective algebraic variety X and a locally free sheaf E, it states:

(X, E) = (-1)ⁱ dim Hⁱ(X, E) = _X ch(E) td(T_X)

where (X, E) is the Euler characteristic of E, ch(E) is the Chern character of E, td(T_X) is the Todd class of the tangent bundle of X, and the integral denotes the evaluation of the top-degree component on the fundamental class of X.

tale Cohomology

tale cohomology, introduced by Alexander Grothendieck and developed with Michael Artin, extends the power of topological cohomology theories to algebraic varieties over arbitrary fields, including positive characteristic fields. This was crucial for the eventual proof of the Weil conjectures by Pierre Deligne.

tale cohomology with coefficients in $Z$⁄/ⁿ$Z$ (or the inverse limit $Z$⁄$Z$) for a prime different from the characteristic of the base field provides a cohomology theory that behaves analogously to singular cohomology with finite coefficients. The -adic cohomology groups Hⁱ(X, $Z$⁄$Z$) projectively carry a continuous action of the absolute Galois group of the base field, making them Galois representations.

Hodge Theory

When working over the complex numbers, algebraic varieties have an underlying complex analytic structure. Hodge theory relates the topological invariants of a smooth projective variety to its algebraic-geometric invariants via the Hodge decomposition.

The Hodge conjecture, one of the Millennium Prize Problems, predicts that for smooth projective varieties over $C$, certain cohomology classes (Hodge classes) are algebraic, i.e., can be represented by linear combinations of algebraic cycles. This deep conjecture has guided much research in algebraic geometry.

Applications

Cohomology theories on algebraic varieties have numerous applications across mathematics:

• Arithmetic geometry: tale cohomology plays a central role in the study of Diophantine equations, leading to Wiles' proof of Fermat's Last Theorem.
• Mirror symmetry: In string theory, cohomological information of Calabi-Yau manifolds relates to their physics via mirror symmetry phenomena.
• Derived categories: The study of derived categories of coherent sheaves has led to new insights in birational geometry and classification theory.
• Moduli spaces: Cohomology of moduli spaces of vector bundles, curves, and other geometric objects provides rich information about their structure.

Conclusion

Cohomology on algebraic varieties continues to be a vibrant area of research. Recent developments include the theory of perfectoid spaces by Peter Scholze, which has led to new perspectives on tale cohomology and p-adic Hodge theory. Additionally, interactions with derived algebraic geometry and homotopy theory have expanded the cohomological toolbox available to algebraic geometers.

The diverse cohomology theories on algebraic varieties form a powerful language that transcends traditional boundaries between different areas of mathematics. Their development represents one of the most profound intellectual achievements in modern mathematics, providing a framework to explore the intricate relationships between geometry, topology, and arithmetic.