The Kernel of the Gysin Homomorphism for Positive Characteristic Fields: Extending Results from Complex Numbers

Core Concepts

This research paper extends the understanding of the Gysin kernel from the realm of complex numbers to uncountable algebraically closed fields of positive characteristic, demonstrating that a theorem about the structure of the Gysin kernel, previously proven over the complex numbers, also holds true in this more general setting.

Abstract

• Bibliographic Information: Schoemann, C., & Werner, S. (2024). The Kernel of the Gysin Homomorphism for Positive characteristic. arXiv preprint arXiv:2411.11417v1.
• Research Objective: The paper aims to prove that a theorem concerning the Gysin kernel, previously established for smooth projective connected surfaces over the complex numbers, remains valid when the underlying field is an uncountable algebraically closed field of positive characteristic.
• Methodology: The authors utilize techniques from algebraic geometry, including the theory of Lefschetz pencils, étale cohomology, and the comparison theorems relating étale cohomology to singular cohomology. They meticulously adapt the arguments used in the complex number case, employing étale base change arguments and lifting to characteristic zero when necessary.
• Key Findings: The paper successfully demonstrates that the Gysin kernel, for a smooth projective connected surface over an uncountable algebraically closed field of positive characteristic, exhibits a structure analogous to the case over the complex numbers. Specifically, it is shown that the Gysin kernel can be expressed as a union of countable translates of an abelian subvariety, and for a very general choice of parameters, this subvariety is either trivial or coincides with a specific abelian subvariety associated with the vanishing cohomology.
• Main Conclusions: The research significantly extends the applicability of a key result about the Gysin kernel to a broader class of fields, encompassing positive characteristic. This generalization provides a deeper understanding of the Gysin homomorphism and its kernel in a more general algebraic setting.
• Significance: The paper contributes to the field of algebraic geometry by demonstrating the robustness of certain geometric constructions and results across different characteristics. It highlights the power of étale cohomology and related techniques in tackling problems that were previously confined to characteristic zero settings.
• Limitations and Future Research: The paper focuses on smooth projective connected surfaces. Exploring the validity of the theorem for higher-dimensional varieties or more general schemes could be a potential avenue for future research. Additionally, investigating the implications of this result for other geometric invariants and constructions would be of interest.

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The Kernel of the Gysin Homomorphism for Positive characteristic

by Claudia Scho... at arxiv.org 11-19-2024

https://arxiv.org/pdf/2411.11417.pdf

Deeper Inquiries

How does the structure of the Gysin kernel change when we move to higher-dimensional varieties or more general schemes?

Moving beyond the setting of smooth projective surfaces over an uncountable algebraically closed field of positive characteristic significantly complicates the study of the Gysin kernel. Here's a breakdown of the challenges and potential changes:
Higher-Dimensional Varieties:

Complexity of Chow Groups: The structure of the Chow group of 0-cycles, CH0(X), becomes vastly more intricate for higher-dimensional varieties. The relatively simple picture we have for curves and surfaces, where we can often relate CH0(X) to Jacobians or Albanese varieties, breaks down.
Failure of Key Correspondences: The nice correspondences between rational, algebraic, and homological equivalence that hold for curves and surfaces may no longer be true. This makes it harder to control the structure of the Gysin kernel using cohomological or geometric methods.
Vanishing Cohomology: The vanishing cohomology, which plays a crucial role in defining the abelian variety Bt, becomes more challenging to compute and understand in higher dimensions.
More General Schemes:

Singularities: Singularities introduce substantial difficulties. The Gysin map itself needs to be defined more carefully, often using derived categories. The structure of the Gysin kernel can be heavily influenced by the singularities of the varieties involved.
Non-Projectivity: For non-projective schemes, many of the tools from intersection theory and the theory of cycles become unavailable or need significant modifications.
Possible Approaches and Modifications:

Higher Chow Groups: Spencer Bloch's theory of higher Chow groups provides a powerful framework for studying algebraic cycles in greater generality. It's plausible that techniques from higher K-theory and motivic cohomology could shed light on the Gysin kernel in more general settings.
Deformations and Specializations: Studying how the Gysin kernel behaves under deformations of varieties or specializations to characteristic p might offer insights.
Derived Categories: Using derived categories and the language of perverse sheaves could provide a more flexible framework for analyzing the Gysin map and its kernel when dealing with singularities.
In summary, while the precise structure of the Gysin kernel in greater generality remains an open question, exploring connections with higher Chow groups, deformation theory, and derived categories offers promising avenues for future research.

Could there be counterexamples to this theorem in the case of countable algebraically closed fields of positive characteristic?

Yes, the countability of the field is crucial for the theorem on the Gysin kernel, and counterexamples are likely to exist over countable algebraically closed fields of positive characteristic.
Here's why countability matters:

Uncountability and Transversality Arguments: The proof of the Gysin kernel theorem relies heavily on what are essentially "dimension-counting" or transversality arguments. These arguments often involve showing that certain "bad" loci within parameter spaces have strictly smaller dimension than the parameter spaces themselves. This allows us to conclude that a "very general" point in the parameter space avoids these bad loci. Over uncountable fields, we have a stronger notion of dimension, making these arguments work.

Countable Fields and "Generic" Behavior: Over countable fields, the notion of a "very general" point becomes much weaker. It's possible to have a countable collection of proper subvarieties that cover the entire parameter space. This means we can no longer guarantee that a "very general" point avoids certain undesirable properties.
Potential Counterexamples:
Constructing explicit counterexamples might be delicate, but the general idea would be to exploit the failure of transversality arguments over countable fields. One could try to construct:

A surface and a linear system where the locus of smooth curves with a "large" Gysin kernel is dense in the parameter space of all curves. In this situation, even a "very general" curve would have a Gysin kernel that is not described by the theorem.
In essence, the countability of the field introduces a level of "rigidity" that can prevent the Gysin kernel from having the nice structure described in the theorem.

What are the implications of this result for understanding the geometry of algebraic cycles and their equivalence relations in positive characteristic?

This result, extending the understanding of the Gysin kernel to positive characteristic, has several important implications for the study of algebraic cycles and their equivalence relations:
1. Deeper Understanding of Chow Groups in Positive Characteristic:

Structure of CH0: The theorem provides a more precise description of the Chow group of 0-cycles of degree zero, CH0(S)deg=0, for surfaces in positive characteristic. It shows that, at least for very general curves on the surface, the kernel of the Gysin map has a well-defined structure related to abelian varieties.
Relationship Between Equivalence Relations: While the proof doesn't directly rely on it, the result provides further evidence that the relationships between rational, algebraic, and homological equivalence for 0-cycles on curves, which are well-understood in characteristic zero, might also hold in positive characteristic.
2. Connections to Other Geometric Constructions:

Albanese and Picard Varieties: The appearance of the Jacobian variety (which is also the Albanese variety for curves) in the description of the Gysin kernel strengthens the link between the geometry of algebraic cycles and these fundamental abelian varieties.
Monodromy Action: The proof's use of Lefschetz pencils and the monodromy action highlights the importance of understanding how families of cycles behave and how this relates to the structure of the Gysin kernel.
3. New Tools and Techniques:

Étale Cohomology: The successful adaptation of the proof from a Hodge-theoretic setting (over the complex numbers) to one using étale cohomology demonstrates the power and flexibility of étale cohomology for studying geometric questions in positive characteristic.
4. Further Research Directions:

Higher Dimensions: The result naturally motivates the question of whether similar structures for the Gysin kernel exist for higher-dimensional varieties in positive characteristic.
Arithmetic Applications: Understanding the Gysin kernel has implications for studying rational points on varieties and other arithmetic questions. The extension to positive characteristic could lead to new insights in arithmetic geometry.
In conclusion, this result represents a significant advance in our understanding of algebraic cycles in positive characteristic. It deepens our knowledge of Chow groups, strengthens connections to other geometric constructions, and opens up new avenues for research in this area.