Hodge conjecture
Introduction
The Hodge conjecture is a central unsolved problem in algebraic geometry, a branch of mathematics that studies the solutions to systems of polynomial equations and their geometric properties. Proposed by the British mathematician William Vallance Douglas Hodge in the mid-20th century, the conjecture is part of the broader framework of the Hodge theory, which relates the topology of a smooth projective algebraic variety to its algebraic cycles. The conjecture is one of the seven Millennium Prize Problems for which the Clay Mathematics Institute has offered a prize of one million dollars for a correct solution.
Background and Context
The Hodge conjecture emerges from the study of complex manifolds and Kähler manifolds, which are special types of differentiable manifolds equipped with additional structure. Hodge theory, developed in the 1930s and 1940s, provides a powerful tool for understanding the relationship between the de Rham cohomology of a manifold and its Dolbeault cohomology. This relationship is encapsulated in the Hodge decomposition, which expresses the cohomology of a Kähler manifold as a direct sum of spaces of harmonic forms.
The conjecture specifically concerns the cohomology classes of a non-singular projective algebraic variety over the complex numbers. It posits that certain cohomology classes, known as Hodge classes, can be represented as rational linear combinations of algebraic cycles. An algebraic cycle is a formal sum of subvarieties of a given dimension, and the conjecture seeks to establish a bridge between the topological and algebraic aspects of these cycles.
Statement of the Hodge Conjecture
Formally, let \( X \) be a smooth projective algebraic variety over the complex numbers, and let \( H^k(X, \mathbb{Q}) \) denote the \( k \)-th cohomology group of \( X \) with rational coefficients. The Hodge conjecture asserts that every Hodge class in \( H^{2p}(X, \mathbb{Q}) \) is a rational linear combination of the classes of algebraic cycles of codimension \( p \).
The conjecture can be restated in terms of the Hodge structure on the cohomology of \( X \). A Hodge class is a cohomology class that lies in the intersection of the \( (p, p) \)-component of the Hodge decomposition with the rational cohomology. The conjecture thus suggests that these classes have a geometric origin, being expressible in terms of algebraic cycles.
Implications and Related Concepts
The Hodge conjecture has profound implications for the understanding of the geometry and topology of algebraic varieties. If true, it would provide a comprehensive framework for understanding the relationship between the topological invariants of a variety and its algebraic structure. The conjecture is closely related to other major conjectures in algebraic geometry, such as the Tate conjecture and the Grothendieck standard conjectures.
The conjecture is also connected to the Lefschetz standard conjecture, which concerns the existence of certain algebraic cycles that induce isomorphisms between cohomology groups. The Hodge conjecture can be viewed as a special case of the generalized Hodge conjecture, which extends the statement to include non-projective varieties and other types of cohomology.
Progress and Partial Results
While the Hodge conjecture remains unsolved in full generality, significant progress has been made in special cases. For example, the conjecture is known to hold for abelian varieties, which are projective varieties with a group structure. The work of Pierre Deligne and others has established the conjecture for certain classes of varieties with additional symmetries.
In dimension two, the conjecture is true for all surfaces, as shown by the Lefschetz theorem on (1,1)-classes. For higher-dimensional varieties, the conjecture has been verified in specific cases, such as complete intersections and certain moduli spaces. However, a general proof or counterexample remains elusive.
Challenges and Open Questions
The difficulty of the Hodge conjecture lies in the complex interplay between the algebraic and topological properties of varieties. One of the main challenges is to construct explicit algebraic cycles representing given Hodge classes, a task that often requires deep insights into the geometry of the variety.
Another challenge is the lack of effective tools for studying the rational cohomology of varieties, particularly in higher dimensions. While Morse theory and other techniques provide powerful methods for analyzing the topology of smooth manifolds, they often fall short in the algebraic setting.
The conjecture also raises fundamental questions about the nature of algebraic cycles and their role in the broader context of algebraic geometry. Understanding the structure of the Chow group of a variety, which classifies algebraic cycles up to rational equivalence, is a key aspect of this inquiry.