Lefschetz ( 1 , 1 ) (1,1) ( 1 , 1 ) -theorem In Positive Characteristic
Introduction to the Lefschetz (1,1)-Theorem
The Lefschetz (1,1)-theorem is a cornerstone result in algebraic geometry, particularly within the realm of complex algebraic varieties. At its core, this theorem provides a bridge between the topological and algebraic nature of a smooth projective variety. It elegantly characterizes the algebraic cohomology classes – those arising from algebraic cycles – within the broader landscape of all cohomology classes. Specifically, it states that a cohomology class of type (1,1) on a complex projective manifold is algebraic if and only if it has integral periods. This profound connection has far-reaching implications, allowing us to use topological tools to study algebraic objects and vice versa.
However, the classical Lefschetz (1,1)-theorem is intrinsically linked to the complex numbers, leveraging the Hodge decomposition and the rich structure of complex cohomology. When we venture into the realm of positive characteristic, where the base field is not the complex numbers but an algebraically closed field of characteristic p > 0, the landscape shifts dramatically. The Hodge decomposition, a fundamental tool in the complex setting, no longer holds in its familiar form. The familiar cohomological machinery built upon singular cohomology gives way to alternative theories, such as étale cohomology and crystalline cohomology, which are better suited to handle the challenges posed by positive characteristic. Therefore, understanding an analogue or a suitable generalization of the Lefschetz (1,1)-theorem in positive characteristic becomes a central question in arithmetic geometry.
In this article, we delve into the intricacies of the Lefschetz (1,1)-theorem in the context of positive characteristic. We will explore the known results, conjectures, and the specific challenges that arise when attempting to translate this classical theorem to fields of characteristic p > 0. Let k be an algebraically closed field of characteristic p > 0, and let W denote its Witt ring. Consider X to be a smooth projective variety defined over k. The central question we address is: What can we say about the image of the cycle class map in this setting? This involves navigating through the complexities of crystalline cohomology, Picard groups, and the unique phenomena that arise in characteristic p.
Background and Challenges in Positive Characteristic
To fully appreciate the subtleties of the Lefschetz (1,1)-theorem in positive characteristic, it's essential to first lay down some groundwork. Let's consider k as an algebraically closed field with a characteristic p > 0. In this setting, the classical tools of complex algebraic geometry, such as the Hodge decomposition, are no longer directly applicable. The Hodge decomposition, which elegantly splits the complex cohomology of a smooth projective variety into subspaces determined by holomorphic and antiholomorphic forms, hinges on the properties of complex numbers. In characteristic p, the de Rham cohomology, which serves as an analogue to singular cohomology, lacks such a decomposition. This absence necessitates the use of alternative cohomological theories that are better suited to handle the nuances of positive characteristic.
One of the primary tools in this domain is étale cohomology. Étale cohomology, developed by Grothendieck, provides a robust framework for studying the topology of algebraic varieties over fields that are not necessarily the complex numbers. It serves as a powerful substitute for singular cohomology and is particularly well-behaved in characteristic p, offering a way to define Betti numbers and study topological invariants. However, even with étale cohomology, certain challenges remain, especially when trying to mirror the Lefschetz (1,1)-theorem. The cycle class map, which associates a cohomology class to an algebraic cycle, behaves differently in characteristic p, and the relationship between algebraic cycles and cohomology classes is more intricate.
Another crucial tool in positive characteristic is crystalline cohomology. Crystalline cohomology, introduced by Grothendieck and further developed by Berthelot, provides a p-adic analogue of de Rham cohomology. It is particularly useful for studying the arithmetic properties of varieties over fields of characteristic p. The crystalline cohomology groups carry a rich structure, including a Frobenius action and a filtration, which makes them a powerful tool for studying algebraic cycles. The Witt ring W of k, which appears in the initial setup, plays a significant role in crystalline cohomology, as it provides the coefficient ring for these cohomology groups. Understanding the interplay between crystalline cohomology and the cycle class map is paramount in formulating and investigating analogues of the Lefschetz (1,1)-theorem.
In addition to these cohomological theories, the Picard group plays a central role. The Picard group, denoted Pic(X), is the group of isomorphism classes of line bundles on the variety X. It is a fundamental invariant in algebraic geometry, encoding much information about the geometry of the variety. The cycle class map often relates the Picard group to specific cohomology groups, and understanding the image of this map is closely tied to the Lefschetz (1,1)-theorem. However, in positive characteristic, the structure of the Picard group and its relationship to cohomology can be quite complex, necessitating careful analysis.
The Brauer group also enters the picture as it is intricately linked to the Picard group and the arithmetic of the variety. The Brauer group measures the obstruction to the local-global principle for certain torsors, and its properties are deeply intertwined with the cohomology of the variety. In positive characteristic, the Brauer group can exhibit behaviors that are not seen in characteristic zero, adding another layer of complexity to the Lefschetz (1,1)-theorem. Understanding the Brauer group and its connection to algebraic cycles is crucial for a comprehensive understanding of the theorem in this setting.
Conjectures and Known Results
Given the challenges and the altered landscape in positive characteristic, the Lefschetz (1,1)-theorem does not have a straightforward analogue. However, several conjectures and known results aim to capture the essence of the theorem in this context. One prominent direction of research focuses on understanding the image of the cycle class map in various cohomology theories. The cycle class map, in general, associates an algebraic cycle (a formal sum of subvarieties) to a cohomology class. The classical Lefschetz (1,1)-theorem essentially characterizes the image of this map in complex cohomology. In positive characteristic, the question becomes: What can we say about the image of the cycle class map in étale or crystalline cohomology?
One approach is to consider the cycle class map
cl: NS(X) ⊗ Z_l → H^2(X, Z_l(1))
where NS(X) denotes the Néron-Severi group of X, l is a prime different from p, and H²(X, Zl(1)) is the étale cohomology group. A significant result in this direction is the Tate conjecture, which posits that the cycle class map is surjective onto the subspace of cohomology classes fixed by a power of the Frobenius endomorphism. The Tate conjecture, while still open in general, provides a deep connection between algebraic cycles and étale cohomology. It suggests that, in a suitable sense, all cohomology classes that are “defined over” the base field k arise from algebraic cycles.
The Tate conjecture has been proven in some specific cases, such as for abelian varieties and certain types of surfaces. However, a general proof remains elusive, making it one of the central open problems in arithmetic geometry. The conjecture is intimately tied to the Lefschetz (1,1)-theorem, as it provides a strong statement about the algebraicity of cohomology classes. If the Tate conjecture holds, it would provide a powerful generalization of the Lefschetz (1,1)-theorem to positive characteristic, albeit in the realm of étale cohomology.
Another avenue of investigation involves crystalline cohomology. The cycle class map in crystalline cohomology takes the form
cl: NS(X) → H^2_{cris}(X/W)
where H²_cris_(X/W) denotes the second crystalline cohomology group of X over the Witt ring W. The structure of crystalline cohomology, with its Frobenius action and Hodge-like filtration, allows for a finer analysis of the cycle class map. In this context, conjectures often focus on the relationship between the image of the cycle class map and the Frobenius action. For instance, one might ask whether the image of the cycle class map is contained in a specific Frobenius eigenspace or whether the crystalline Chern classes of algebraic cycles satisfy certain integrality conditions.
One important result in this direction is the Bloch-Kato conjecture (now a theorem, thanks to work by Rost, Voevodsky, and others), which has implications for the cycle class map in crystalline cohomology. The Bloch-Kato conjecture relates Milnor K-theory to Galois cohomology and, in the context of algebraic geometry, provides insights into the structure of motivic cohomology. Motivic cohomology is a powerful generalization of Chow groups and is believed to be the universal cohomology theory for algebraic varieties. The Bloch-Kato conjecture, by linking motivic cohomology to more accessible cohomology theories, provides tools for studying the image of the cycle class map and the algebraicity of cohomology classes.
Furthermore, there are results and conjectures concerning the p-adic Picard scheme and its relationship to crystalline cohomology. The p-adic Picard scheme is a p-adic analogue of the Picard variety, and it plays a crucial role in understanding the arithmetic of line bundles in positive characteristic. The crystalline Chern class map relates the p-adic Picard scheme to crystalline cohomology, and understanding the image of this map is essential for a p-adic Lefschetz (1,1)-theorem. Conjectures in this area often involve the crystalline Chern class map being an isomorphism onto a specific subspace of crystalline cohomology, thereby providing a precise description of the algebraic cohomology classes.
Specific Challenges and Open Questions
Despite the progress made in understanding the Lefschetz (1,1)-theorem in positive characteristic, several challenges and open questions remain. These challenges stem from the unique phenomena that arise in characteristic p and the limitations of existing cohomological tools. One of the main challenges is the absence of a universal cohomology theory that captures all the desired properties of singular cohomology in characteristic zero. Étale cohomology and crystalline cohomology each have their strengths, but neither fully replicates the behavior of singular cohomology. This necessitates a more nuanced approach, often involving the interplay between different cohomology theories.
One open question revolves around the precise relationship between algebraic cycles and crystalline cohomology classes. While the cycle class map provides a link between them, the image of this map is not fully understood. Specifically, it is not clear whether all crystalline cohomology classes that “look algebraic” (in some appropriate sense) actually arise from algebraic cycles. This question is closely tied to the Tate conjecture, but it also involves the fine structure of crystalline cohomology and the Frobenius action. Understanding the crystalline Chern classes of algebraic cycles and their integrality properties remains a central problem.
Another challenge lies in the behavior of the Picard group in positive characteristic. The Picard group, which classifies line bundles, is a fundamental invariant in algebraic geometry. However, in characteristic p, the Picard group can exhibit behaviors that are not seen in characteristic zero. For example, the Picard scheme, which represents the Picard group, can be non-reduced in characteristic p, leading to complexities in its structure and its relationship to cohomology. Understanding the p-adic Picard scheme and its connection to crystalline cohomology is crucial for a comprehensive p-adic Lefschetz (1,1)-theorem.
The Hodge-Witt cohomology offers a potential avenue for further investigation. Hodge-Witt cohomology, developed by Illusie and Raynaud, provides a refinement of crystalline cohomology that captures finer arithmetic information. It interpolates between crystalline cohomology and de Rham-Witt cohomology and offers a more detailed picture of the cohomology of varieties in characteristic p. Exploring the cycle class map in Hodge-Witt cohomology and its relationship to algebraic cycles may lead to new insights into the Lefschetz (1,1)-theorem.
Furthermore, the study of specific classes of varieties, such as abelian varieties, K3 surfaces, and Calabi-Yau manifolds, provides valuable test cases for conjectures and results concerning the Lefschetz (1,1)-theorem. These varieties often exhibit special properties that make them more amenable to analysis, and results in these cases can shed light on the general picture. For example, the Tate conjecture has been proven for abelian varieties, and the study of K3 surfaces in characteristic p has revealed interesting phenomena related to the Picard group and crystalline cohomology.
In conclusion, the Lefschetz (1,1)-theorem in positive characteristic remains an active area of research with many open questions and challenges. While the classical theorem does not have a direct analogue, ongoing work in étale cohomology, crystalline cohomology, and Hodge-Witt cohomology provides a rich framework for understanding the relationship between algebraic cycles and cohomology classes in this setting. Conjectures such as the Tate conjecture and results concerning the Bloch-Kato conjecture offer promising directions for future research, and the study of specific classes of varieties continues to provide valuable insights. The quest for a comprehensive p-adic Lefschetz (1,1)-theorem remains a central goal in arithmetic geometry, promising to deepen our understanding of the interplay between algebra, topology, and arithmetic in positive characteristic.