The semiregularity theorem for equivariant noncommutative varieties

Abstract: We generalize the classical semiregularity theorem of Buchweitz and Flenner to the setting of noncommutative algebraic geometry, with group actions. This applies in particular to twisted derived categories, in which case it answers a question of Markman and streamlines part of his proof of the Hodge conjecture for abelian fourfolds. Along the way, we prove that for many finite group actions on derived categories of varieties, the invariant category is of geometric origin.

• The paper generalizes the Buchweitz-Flenner semiregularity theorem to twisted and equivariant contexts, proving unobstructed deformations via an injective semiregularity map.
• The paper develops an equivariant formalism distinguishing G-semi-regularity from weak G-semi-regularity, ensuring smooth moduli for objects under group actions.
• The paper unifies deformation theory with Hodge-theoretic applications, streamlining approaches to the algebraicity of Hodge classes in noncommutative settings.

Semiregularity for Equivariant Noncommutative Varieties

Introduction and Motivation

This paper establishes a categorical generalization of the classical semiregularity theorem for deformations of objects in algebraic geometry, extending it to the context of noncommutative varieties with symmetry, specifically, to semiorthogonal components of derived categories equipped with compatible linear group actions. The author develops the corresponding formalism needed for equivariant derived categories and provides results pertinent to applications in derived invariants, integral Hodge theory, and moduli of objects in families with group actions. The framework unifies and extends previous results, notably answering and simplifying technical aspects in E. Markman's proofs of the Hodge conjecture for abelian fourfolds.

Statement and Generalization of Semiregularity

The main theorems of the paper provide, in particular, the following:

• A generalization of the Buchweitz-Flenner semiregularity theorem to twisted and equivariant settings, replacing the classical role of commutative varieties by semiorthogonal components of derived categories, and allowing for perfect complexes (possibly twisted by Brauer classes) as objects under consideration.
• A formulation and proof of an equivariant semiregularity theorem for objects equipped with a finite group action. This introduces—and distinguishes between—$G$-semiregularity (injectivity of the semiregularity map on $G$-invariant subspaces) and weak $G$-semiregularity (semiregularity in the invariant category) and proves that, up to mild hypotheses, the latter suffices for the smoothness and deformation-theoretic conclusions characteristic of classical semiregularity.

A key statement in the context of abelian varieties (see Theorem 1.2) is:

Given a family $X\to S$ of abelian varieties and a perfect complex $E_0$ on the special fiber $X_0$ that is weakly $G$-semiregular and simple, if its Mukai vector (possibly twisted) remains Hodge along $S$, then $E_0$ deforms (up to a Brauer-twist) and the Hodge-theoretic condition persists as algebraicity.

This thereby allows for the verification of algebraicity of Hodge classes via deformation-theoretic techniques in equivariant, noncommutative, and twisted settings.

Technical Framework

Hochschild Homology and the Semiregularity Morphism

The deformation theory of objects in a linear category $C$ (over a base scheme $G$0) is governed by Ext groups, and the obstruction to deformations lies in $G$1. The semiregularity morphism, a map $G$2, is shown to play the same role as in commutative geometry: injectivity on this map implies that $G$3 is unobstructed in the moduli problem.

The construction is made functorial for group actions and for semiorthogonal components by careful consideration of linear enhancements, Hochschild homology with coefficients, and appropriate base change.

Equivariant Formalism

The author provides a categorical language for finite group actions on triangulated categories (and their semiorthogonal pieces) adequate for studying invariance and descent. The main technical distinction is between:

• $G$4-semiregularity: injectivity of the semiregularity morphism on $G$5-invariant Exts.
• Weak $G$6-semiregularity: semiregularity in the equivariant/invariant category.

A key lemma asserts that, in characteristic zero and for $G$7 invertible, $G$8-semiregularity implies weak $G$9-semiregularity.

Geometricity of Invariant Categories

A technical challenge is demonstrating when invariant categories of group actions on derived categories are themselves of geometric origin. The paper proves:

• For many group actions (notably, those acting via the identity component of automorphisms and tensoring by line bundles, as in abelian varieties), invariants can be realized as semiorthogonal components of derived categories of stacks or (twisted) gerbe-augmented varieties.
• This geometricity is crucial for lifting deformation-theoretic results and accessing the full Hodge-theoretic formalism in the equivariant context.

Hodge Theory and Variational Deformation

A central theme is the compatibility of categorical K-theory and Hochschild-theoretic invariants with variation of Hodge structures in families. The author extends Hotchkiss's and Perry's previous work to the setting of twisted and equivariant derived categories, describing:

• How rational (and, in certain cases, integral) Hodge classes are transported in families, including for twisted perfect complexes.
• The precise structure of algebraic realizations and their behavior under group actions.

The rationality statements are essential for comparison with the Hodge conjecture and for showing the algebraicity of limit classes in deformations.

Applications and Implications

Streamlining Arguments for the Hodge Conjecture

One of the core achievements is clarifying and strengthening the technical underpinnings of Markman's recent work on the Hodge conjecture for special classes of abelian varieties. Where Markman relied on involved ad hoc arguments to descend semiregularity or algebraicity properties through derived equivalences and group symmetries, this paper supplies a uniform categorical formalism that both generalizes and simplifies those steps.

Moduli Smoothness Results

A principal technical result is the proof of smoothness for moduli spaces of objects in noncommutative and equivariant settings at semiregular points—a major extension of classical results, going beyond the CY2 case with a categorical approach.

Equivariant Techniques for Obstruction Vanishing

The introduction of the equivariant semiregularity theorem has consequences for the computation and vanishing of obstruction spaces in equivariant deformation theory, with further ramifications for constructing and understanding moduli of objects with symmetries.

Future Directions

The categorical and equivariant framework presented signals potential advances in:

• The development of noncommutative Hodge theory and integral Hodge conjecture in more general settings, including for families of noncommutative varieties and derived categories beyond the case of schemes.
• The explicit description of invariants and fixed-point categories under group actions on derived categories, particularly those arising from automorphism groups of Calabi-Yau and abelian varieties.
• The study of derived categories of stacks and their semiorthogonal components in both algebraic and arithmetic directions, leveraging equivariant techniques for the study of moduli, cycles, and period maps.

Conclusion

This paper provides a comprehensive generalization of the semiregularity theorem to the field of equivariant, noncommutative, and twisted derived categories, establishing both the technical machinery and the categorical deformation-theoretic results required to handle moduli and Hodge-theoretic questions in this context. The methods and results lay foundational work for future research in deformation theory, noncommutative geometry, and the categorical aspects of algebraic cycles and Hodge theory, opening avenues for more uniform approaches to equivariant problems in algebraic geometry and beyond.

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Reference: "The semiregularity theorem for equivariant noncommutative varieties" (2604.00511)