Variation of Archimedean Zeta Function and the n/d-Conjecture for Generic Multiplicities

Core Concepts

This research paper introduces a variation of the Archimedean zeta function and investigates its properties, particularly the existence of poles, to prove the n/d-conjecture for generic multiplicities in the context of hyperplane arrangements.

Abstract

• Bibliographic Information: Shi, Q., & Zuo, H. (2024). Variation of Archimedean Zeta Function and n/d-Conjecture for Generic Multiplicities. arXiv:2411.00757v1 [math.AG].

• Research Objective: The paper aims to explore the properties of a newly introduced variation of the Archimedean zeta function and apply these findings to prove the n/d-conjecture for generic multiplicities, a conjecture related to the Bernstein-Sato polynomials of hyperplane arrangements.

• Methodology: The authors define a variation of the Archimedean zeta function, denoted as Zϕ(b, s), which depends on a tuple of multiplicities (b1, ..., br) and a complex variable s. They analyze the meromorphic properties of this function, particularly the existence and behavior of its poles, using techniques from complex analysis and algebraic geometry, including log resolutions and integration by parts.

• Key Findings: The paper establishes the meromorphic nature of the variation of the Archimedean zeta function on a specific domain. It introduces the concept of a "good tuple" of multiplicities and demonstrates that the existence of a good tuple at a specific divisor implies the existence of a pole for the variation of the zeta function. This result is then used to prove the n/d-conjecture for generic multiplicities, meaning the conjecture holds for a dense open subset of multiplicities. Additionally, the paper provides a new proof of the n/d-conjecture in dimension 2 without requiring the generic multiplicity condition.

• Main Conclusions: The study successfully introduces and analyzes a variation of the Archimedean zeta function, demonstrating its utility in proving the n/d-conjecture for generic multiplicities. This result contributes significantly to the understanding of Bernstein-Sato polynomials and their connection to the geometry of hyperplane arrangements. The authors also link their findings to the strong monodromy conjecture, showing its validity for hyperplane arrangements with generic multiplicities.

• Significance: This research significantly advances the field of singularity theory, particularly the study of hyperplane arrangements and their associated invariants. The introduction and analysis of the variation of the Archimedean zeta function provide a new tool for investigating the n/d-conjecture and the broader monodromy conjecture, which have implications for various areas of mathematics.

• Limitations and Future Research: The paper primarily focuses on hyperplane arrangements, leaving the exploration of the variation of the Archimedean zeta function and the n/d-conjecture for more general hypersurfaces as a potential avenue for future research. Further investigation into the properties of "good tuples" and their implications for the poles of the zeta function could yield additional insights.

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Variation of Archimedean Zeta Function and \texorpdfstring{$n/d$}{n/d}-Conjecture for Generic Multiplicities

by Quan Shi, Hu... at arxiv.org 11-04-2024

https://arxiv.org/pdf/2411.00757.pdf

Deeper Inquiries

How can the variation of the Archimedean zeta function be applied to study other geometric or topological invariants beyond the Bernstein-Sato polynomial?

The variation of the Archimedean zeta function, as introduced in the paper, provides a powerful tool to study the behavior of geometric and topological invariants in families of hypersurfaces with varying multiplicities. While the authors primarily focus on its application to the Bernstein-Sato polynomial and the n/d-conjecture, its potential extends beyond these areas. Here are some potential avenues for further exploration:

Jumping Numbers and Multiplier Ideals: As mentioned in Remark 2.2, the roots of the Bernstein-Sato polynomial are closely related to the jumping numbers and multiplier ideals associated with a polynomial. The variation of the Archimedean zeta function could offer insights into how these invariants change as the multiplicities of the defining equations vary. This could lead to a deeper understanding of the geometry of singularities in families.

Milnor Fibrations and Monodromy: The topology of a hypersurface singularity is captured by its Milnor fibration and the associated monodromy operator. The variation of the Archimedean zeta function could potentially be used to study the behavior of these topological invariants in families. For instance, one could investigate how the eigenvalues of the monodromy operator, which are related to the roots of the Bernstein-Sato polynomial by the Kashiwara-Malgrange theorem, vary with the multiplicities.

Hodge Theory and Mixed Hodge Structures: The Archimedean zeta function has connections to Hodge theory, particularly through the study of mixed Hodge structures on the cohomology of the Milnor fiber. The variation of the zeta function could potentially be used to study the variation of mixed Hodge structures in families of hypersurfaces, leading to a finer understanding of their Hodge-theoretic properties.

Motivic Invariants: The motivic zeta function, a more sophisticated invariant than the Archimedean zeta function, encodes richer information about the geometry of a variety. The variation of the Archimedean zeta function could potentially provide insights into the variation of motivic invariants in families, leading to a deeper understanding of their motivic nature.
These are just a few potential directions for future research. The key takeaway is that the variation of the Archimedean zeta function provides a new lens through which to study the geometry and topology of hypersurface singularities in families, with the potential to uncover new connections and insights beyond the scope of the current paper.

Could there be alternative approaches to proving the n/d-conjecture that do not rely on the concept of "good tuples" or the variation of the Archimedean zeta function?

While the concept of "good tuples" and the variation of the Archimedean zeta function provide a powerful framework for approaching the n/d-conjecture, alternative approaches could potentially offer different insights and circumvent the limitations of the current method. Here are some possibilities:

Direct Analysis of D-Modules: The Bernstein-Sato polynomial is fundamentally an invariant arising from the theory of D-modules. A deeper analysis of the D-module associated with a hyperplane arrangement, perhaps leveraging its combinatorial structure, could potentially lead to a direct proof of the n/d-conjecture without relying on zeta functions.

Combinatorial Methods: Hyperplane arrangements possess a rich combinatorial structure encoded in their intersection lattice. It might be possible to exploit this combinatorial structure to prove the n/d-conjecture through purely combinatorial arguments. This approach could potentially lead to a more conceptual understanding of the conjecture and its relation to the combinatorics of hyperplane arrangements.

Resolution of Singularities and Hodge Theory: The canonical log resolution of a hyperplane arrangement plays a crucial role in the current approach. A more refined analysis of this resolution, perhaps using tools from Hodge theory or motivic integration, could potentially lead to a proof of the n/d-conjecture that bypasses the need for "good tuples" or zeta functions.

Deformation Theory: One could try to deform a given hyperplane arrangement to a simpler one for which the n/d-conjecture is known to hold. By carefully analyzing how the Bernstein-Sato polynomial changes under such deformations, it might be possible to deduce the conjecture for the original arrangement.
It's important to note that the n/d-conjecture has remained open for a significant time, suggesting that a proof might require novel ideas or a deeper understanding of the interplay between the algebraic, geometric, and combinatorial aspects of hyperplane arrangements.

What are the implications of the connection between the n/d-conjecture and the monodromy conjecture for other areas of mathematics, such as number theory or representation theory?

The connection between the n/d-conjecture and the monodromy conjecture hints at a deep relationship between the singularities of algebraic varieties and their topological and arithmetic properties. This connection has potential implications for other areas of mathematics, including:

Number Theory and Arithmetic Geometry: The monodromy conjecture, particularly in its p-adic formulation, has profound implications for the study of L-functions and the Riemann Hypothesis. A proof of the n/d-conjecture, and consequently the monodromy conjecture for hyperplane arrangements, would provide further evidence for these conjectures and potentially offer new tools for tackling them. Moreover, hyperplane arrangements arise naturally in the study of moduli spaces and other arithmetic objects, so understanding their singularities could shed light on their arithmetic properties.

Representation Theory: The monodromy operator associated with a singularity can be viewed as a representation of the fundamental group of the complement of the singular locus. The n/d-conjecture, by implying the monodromy conjecture, would provide information about the structure of these monodromy representations for hyperplane arrangements. This could have implications for the representation theory of braid groups and other related groups.

Mirror Symmetry: Mirror symmetry, a duality between symplectic and complex geometry, predicts surprising connections between the geometry of Calabi-Yau manifolds and their "mirror" partners. Hyperplane arrangements, through their connection to toric varieties, play a role in mirror symmetry. Understanding their singularities and monodromy properties could provide insights into the mirror symmetry phenomenon.

Singularity Theory and Algebraic Geometry: The n/d-conjecture and the monodromy conjecture are fundamental questions in singularity theory. A proof of these conjectures would represent a significant advance in the field, potentially leading to new techniques and insights applicable to a broader class of singularities.
The connection between the n/d-conjecture and the monodromy conjecture highlights the interconnectedness of seemingly disparate areas of mathematics. Further exploration of this connection promises to yield fruitful results and deepen our understanding of the interplay between algebra, geometry, topology, and number theory.