The Semiregularity Map as a Tangent of the Generalized Abel-Jacobi Map on Derived Moduli Stacks

Core Concepts

The semiregularity map, which measures obstructions to deforming perfect complexes and the failure of Chern characters to remain Hodge classes under deformation, can be realized as the tangent of a generalized Abel-Jacobi map on the derived moduli stack of perfect complexes.

Abstract

Bibliographic Information:

Pridham, J.P. (2024). Semiregularity as a consequence of Goodwillie's theorem. arXiv preprint arXiv:1208.3111v4.

Research Objective:

This paper aims to establish a connection between the semiregularity map and the generalized Abel-Jacobi map in the context of derived deformation theory. The author seeks to prove and generalize the semiregularity conjectures proposed by Bloch and later extended by Buchweitz and Flenner.

Methodology:

The author utilizes tools from derived algebraic geometry, particularly derived deformation theory and cyclic homology. The key insight is the use of Goodwillie's theorem on nilpotent ideals, which allows for the replacement of rational Betti cohomology with Hartshorne's algebraic de Rham cohomology in the definition of Deligne cohomology.

Key Findings:

• The semiregularity map can be realized as the tangent of a generalized Abel-Jacobi map on the derived moduli stack of perfect complexes.
• This realization proves and generalizes the semiregularity conjectures, demonstrating that the semiregularity map measures the failure of the Chern character to remain a Hodge class under deformation.
• The results hold for smooth and singular varieties, derived stacks, and even certain non-commutative spaces.

Main Conclusions:

The paper provides a new perspective on the semiregularity map, connecting it to the well-established theory of Abel-Jacobi maps. This connection has significant implications for understanding obstructions in deformation theory and constructing reduced obstruction theories in enumerative geometry.

Significance:

This research contributes significantly to the fields of algebraic geometry and derived algebraic geometry. It provides a powerful tool for studying deformation theory and has applications in enumerative geometry, particularly in constructing reduced obstruction theories for Gromov-Witten and Pandharipande-Thomas invariants.

Limitations and Future Research:

The paper primarily focuses on theoretical aspects of the semiregularity map. Further research could explore the computational aspects and applications of these results in specific geometric contexts. Additionally, investigating the connection between the semiregularity map and other invariants in derived algebraic geometry could lead to new insights.

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Key Insights Distilled From

Semiregularity as a consequence of Goodwillie's theorem

by J. P. Pridha... at arxiv.org 11-06-2024

https://arxiv.org/pdf/1208.3111.pdf

Deeper Inquiries

How can the computational aspects of the generalized Abel-Jacobi map be utilized to calculate semiregularity maps explicitly in specific geometric settings?

Answer:
While the paper establishes a powerful theoretical connection between the generalized Abel-Jacobi map and semiregularity, leveraging it for explicit computations can be challenging. Here's a breakdown of the involved aspects and potential approaches:
Challenges:

Abstract Nature of the Construction: The Abel-Jacobi map is defined using sophisticated tools like ∞-categories and derived deformation theory. Directly working with these objects can be computationally demanding.
Cyclic Homology Computations: The target of the Abel-Jacobi map involves cyclic homology, which, while computable using tools like Hochschild-Kostant-Rosenberg (HKR) isomorphisms and spectral sequences, can still be complex for specific varieties.
Explicit Representatives: Extracting concrete information about the semiregularity map requires understanding explicit representatives of elements in cyclic homology and relating them to the Lefschetz map.
Potential Approaches:

Specific Geometric Settings: Focusing on cases with simpler geometry, like curves, surfaces, or complete intersections, can make the computations more tractable. For example, on a smooth projective curve, the HKR isomorphism provides a direct link between cyclic homology and differential forms, simplifying the analysis.
Exploiting Duality and Degenerations: Techniques from Hodge theory, like duality and studying degenerations of varieties, can provide insights into the structure of the relevant cohomology groups and potentially simplify calculations.
Comparison with Existing Methods: Relating the Abel-Jacobi approach to more classical methods for computing semiregularity, like those using Atiyah classes and traces, might offer computational advantages in certain situations.
Overall:
Explicitly calculating semiregularity maps via the generalized Abel-Jacobi map remains an open area for exploration. Progress likely requires a combination of theoretical insights, computational tools, and focusing on specific geometric contexts where the involved structures become more manageable.

Could there be alternative geometric interpretations of the semiregularity map beyond its connection to the Abel-Jacobi map?

Answer:
Yes, exploring alternative geometric interpretations of the semiregularity map is a promising avenue for research. Here are some potential directions:
1. Deformation of Hodge Structures:

Beyond First-Order: The current connection to the Abel-Jacobi map primarily captures first-order deformations. Investigating higher-order deformations of Hodge structures associated with the complex or the variety could reveal deeper connections to semiregularity.
Variations of Hodge Structures: Studying how the semiregularity map behaves under variations of Hodge structures, particularly near singular points in the moduli space, might offer new geometric insights.
2. Characteristic Classes and Geometric Obstructions:

Chern-Simons Theory: As hinted at in the context, exploring connections to Chern-Simons theory, which provides a geometric framework for understanding secondary characteristic classes, could be fruitful.
Obstruction Bundles: Interpreting the semiregularity map in terms of obstruction bundles or sheaves associated with deformation problems might provide a more direct geometric understanding.
3. Derived Geometry and Categorical Approaches:

Derived Categories and Fourier-Mukai Transforms: Investigating the behavior of the semiregularity map under derived equivalences, potentially using tools like Fourier-Mukai transforms, could reveal new connections to the derived geometry of the variety.
Non-Commutative Motives: Since semiregularity has implications for non-commutative geometry, exploring its interpretation in the context of non-commutative motives might offer a broader perspective.
Overall:
The connection between the semiregularity map and the Abel-Jacobi map provides a valuable starting point. However, exploring alternative geometric interpretations, particularly those rooted in Hodge theory, characteristic classes, and derived geometry, holds the potential to uncover a richer and more nuanced understanding of this important invariant.

How does the concept of semiregularity extend to other areas of mathematics where obstruction theories and deformation problems arise, such as mirror symmetry or non-commutative geometry?

Answer:
The concept of semiregularity, with its interplay of obstruction theories and deformation problems, naturally extends to other areas of mathematics. Here's a glimpse into its potential roles in mirror symmetry and non-commutative geometry:
1. Mirror Symmetry:

Homological Mirror Symmetry: Semiregularity could play a role in understanding how the derived categories of coherent sheaves on mirror manifolds are related. Obstructions to deforming objects and the corresponding semiregularity maps might have interesting counterparts on the mirror side.
Syzgy Varieties and Fukaya Categories: Mirror symmetry often relates algebraic varieties to symplectic manifolds. Investigating semiregularity in the context of Lagrangian submanifolds and the Fukaya category, which governs their deformations, could be insightful.
Quantum Corrections: Mirror symmetry often involves "quantum corrections" to classical geometric notions. Understanding how semiregularity behaves under these corrections could reveal new aspects of the theory.
2. Non-Commutative Geometry:

Deformation Quantization: Semiregularity could provide tools for studying deformation quantization, where one deforms a commutative algebra into a non-commutative one. Obstructions to such deformations and their relation to semiregularity maps could be fruitful to explore.
Non-Commutative Hodge Theory: Developing a robust notion of semiregularity in non-commutative Hodge theory, building on the connections to cyclic homology, could provide new invariants for non-commutative spaces.
Derived Algebraic Geometry: The framework of derived algebraic geometry provides a natural setting for studying deformation problems. Investigating semiregularity in this context, particularly for derived stacks and non-commutative spaces, could lead to new insights.
Overall:
The presence of obstruction theories and deformation problems in mirror symmetry and non-commutative geometry suggests that the concept of semiregularity, appropriately adapted, could play a significant role in these areas. Exploring these connections has the potential to enrich our understanding of both semiregularity itself and the fields to which it is applied.