Twisted Coherent Sheaves in Algebraic Geometry

• Twisted coherent sheaves are coherent sheaves modified by a Brauer class via gerbe data, allowing for obstructions to global triviality.
• They facilitate the construction of twisted Chern characters, Mukai vectors, and intersection products, crucial for moduli spaces and enumerative invariants.
• They underpin derived equivalences and Fourier–Mukai transforms, offering a robust framework for noncommutative and deformation-theoretic studies.

A twisted coherent sheaf is a generalization of the concept of a coherent sheaf, designed to encode the obstruction to the existence of global vector bundles or complexes on a space $X$ arising from nontrivial elements in the Brauer group $\mathrm{Br}(X)$. This obstruction is classically parametrized by a gerbe, leading to the notion of $\alpha$-twisted sheaves for a Brauer class $\alpha \in \mathrm{Br}(X)$. Twisted coherent sheaves play a central role across derived algebraic geometry, noncommutative geometry, and moduli theory, underpinning the structure of twisted derived categories, moduli spaces, and new geometric invariants in both arithmetic and enumerative contexts.

1. Formal Definitions and Basic Properties

Let $X$ be a scheme or smooth projective variety, and $\alpha \in \mathrm{Br}(X)$ a (torsion) Brauer class. There are several equivalent definitions of an $\alpha$-twisted coherent sheaf:

• Gerbe-theoretic: Let $\mathcal{X}_\alpha \to X$ be a $G_m$- or $\mu_n$-gerbe with band determined by $\mathrm{Br}(X)$0. An $\mathrm{Br}(X)$1-twisted coherent sheaf is a coherent $\mathrm{Br}(X)$2-module $\mathrm{Br}(X)$3 for which the $\mathrm{Br}(X)$4-action on $\mathrm{Br}(X)$5 is via the tautological character (Camere et al., 2017, Lane, 2024, Jiang, 2019). When $\mathrm{Br}(X)$6 is covered by étale charts $\mathrm{Br}(X)$7, $\mathrm{Br}(X)$8 determines a cocycle $\mathrm{Br}(X)$9, and a twisted sheaf is a collection $\alpha$0 with

$\alpha$1

such that

$\alpha$2

• Azumaya algebra formulation: If $\alpha$3 is an Azumaya algebra representing $\alpha$4, then the abelian category of $\alpha$5-twisted sheaves is equivalent to the category $\alpha$6 of coherent $\alpha$7-modules (Antieau, 2013).
• Derived categories: $\alpha$8 denotes the bounded derived category of $\alpha$9-twisted coherent sheaves (Lane, 2024).

Twisted sheaves are locally isomorphic to vector bundles or coherent sheaves, but the gluing data is modified by $\alpha \in \mathrm{Br}(X)$0, obstructing global triviality unless $\alpha \in \mathrm{Br}(X)$1.

2. Twisted Chern Characters, Mukai Vectors, and Intersection Theory

Twisted sheaves permit the construction of appropriate characteristic classes and intersection theory:

• Twisted Chern character: Given a $\alpha \in \mathrm{Br}(X)$2-field lift $\alpha \in \mathrm{Br}(X)$3 of $\alpha \in \mathrm{Br}(X)$4 (where $\alpha \in \mathrm{Br}(X)$5), the twisted Chern character of a twisted sheaf $\alpha \in \mathrm{Br}(X)$6 is defined by

$\alpha \in \mathrm{Br}(X)$7

in $\alpha \in \mathrm{Br}(X)$8 (Camere et al., 2017, Jiang, 2019).

• Twisted Mukai vector: For K3 surfaces or holomorphic symplectic varieties, the (twisted) Mukai vector is

$\alpha \in \mathrm{Br}(X)$9

with a natural bilinear Mukai pairing

$X$0

This formalism governs the topology and deformation theory of twisted sheaf moduli (Camere et al., 2017, Jiang, 2019).

• Intersection numbers and Riemann–Roch: The Hirzebruch–Riemann–Roch formula for twisted sheaves involves the above Mukai vector and pairing, e.g.,

$X$1

3. Moduli Spaces and Stability for Twisted Sheaves

Twisted coherent sheaves admit natural notions of slope and Gieseker stability, extended via the $X$2-field or equivalently via the gerbe data:

• Slope stability: For a polarization $X$3 and B-field $X$4, the $X$5-twisted slope is

$X$6

Twisted sheaf $X$7 is $X$8-(semi)stable if any nontrivial $X$9-twisted subsheaf $\alpha \in \mathrm{Br}(X)$0 satisfies $\alpha \in \mathrm{Br}(X)$1 (Camere et al., 2017, Jiang, 2019).

• Moduli stacks/spaces: There is a projective moduli stack $\alpha \in \mathrm{Br}(X)$2 parameterizing semistable $\alpha \in \mathrm{Br}(X)$3-twisted sheaves of fixed data; under stability, this is a μ_n-gerbe over the coarse moduli scheme, a projective variety (Camere et al., 2017, Jiang, 2019). Their expected dimension matches the virtual dimension calculated via deformation theory and Riemann–Roch.
• Applications: On K3 surfaces, moduli spaces $\alpha \in \mathrm{Br}(X)$4 of twisted sheaves with fixed Mukai vector are projective irreducible holomorphic symplectic varieties, deformation-equivalent to Hilbert schemes $\alpha \in \mathrm{Br}(X)$5 whenever $\alpha \in \mathrm{Br}(X)$6 (Camere et al., 2017). For surfaces, moduli of twisted sheaves are used to define enumerative invariants (e.g., twisted Vafa–Witten invariants) and to study rationality/unirationality questions (Jiang, 2019, Camere et al., 2017).

4. Twisted Derived Categories, Fourier–Mukai, and Morita Theory

The abelian category of twisted sheaves forms the basis for the twisted derived category $\alpha \in \mathrm{Br}(X)$7. Fundamental results include:

• Fourier–Mukai theory: Twisted derived equivalences generalize classical Fourier–Mukai equivalences. If $\alpha \in \mathrm{Br}(X)$8 and $\alpha \in \mathrm{Br}(X)$9 are torsors under abelian varieties and $\alpha$0 for a Brauer class $\alpha$1, then $\alpha$2 is a fine moduli space of simple semi-homogeneous sheaves on $\alpha$3 and the equivalence is realized by a universal (twisted) Poincaré bundle (Lane, 2024).
• Tensoring and push-forwards: The derived tensor product of an $\alpha$4- and $\alpha$5-twisted sheaf yields an $\alpha$6-twisted complex. The pull-back and push-forward functors in the derived category are defined via the total gerbe (Lane, 2024).
• Morita theory and reconstruction theorem: Equivalence of the abelian categories of $\alpha$7-twisted quasi-coherent sheaves on $\alpha$8 and $\alpha$9-twisted sheaves on $\mathcal{X}_\alpha \to X$0 implies an isomorphism $\mathcal{X}_\alpha \to X$1 and $\mathcal{X}_\alpha \to X$2 in the Brauer group (Antieau, 2013). This connects to Toën's theory of derived Azumaya algebras and the classification of stacks of twisted perfect complexes.

5. Twisted Sheaves, Stacks, and K-Theory

Twisted coherent sheaves naturally extend to stacks and higher categorical settings:

• Simplicial presheaves and cochain data: Twisted sheaves can be described via Maurer–Cartan descent data or twisting cochains on open covers, with higher homotopy and homological structures (as in the model of Toledo–Tong, O'Brian, Green, and subsequent generalization to stacks) (Hosgood et al., 2023). The descent equation,

$\mathcal{X}_\alpha \to X$3

encodes the obstruction and gluing data.

• Extension to stacks: All definitions and descent formulations transfer directly to Deligne–Mumford and Artin stacks, with twisted $\mathcal{X}_\alpha \to X$4-theory, pushforwards, and Riemann–Roch formulas realized in the stack-theoretic setting (Hosgood et al., 2023).

6. Twisted Sheaves in Advanced Geometric Constructions

Twisted coherent sheaves underlie a variety of advanced geometric and physical applications:

• Derived categories and Satake categories: In geometric representation theory, one has equivalences between derived categories of twisted (monodromic) perverse sheaves and coherent sheaves on a twisted version of the Langlands dual Lie algebra, as in the context of the twisted Satake equivalence (Singh, 2012).
• Arithmetic and enumerative invariants: For $\mathcal{X}_\alpha \to X$5-torsion gerbes, the moduli spaces of $\mathcal{X}_\alpha \to X$6-twisted sheaves/higgs sheaves are central to the formulation of Vafa–Witten invariants, their twisted analogues, and S-duality conjectures (Jiang, 2019). Virtual fundamental classes and deformation-obstruction theory for twisted sheaves mirror the untwisted case in structure.
• Noncommutative and deformation-theoretic geometry: Twisted sheaves capture the structure of Azumaya algebras and their deformations, providing modular examples (e.g., rigid hyperholomorphic sheaves over deformations of K3 surfaces) whose structure and Brauer class are rigid under certain deformations (Markman et al., 2017).

7. Twisted Sheaves on Abelian Varieties and $\mathcal{X}_\alpha \to X$7-Twisted Sheaves

On abelian varieties, the categorification of twisting extends to $\mathcal{X}_\alpha \to X$8-twisted sheaves and is closely intertwined with the geometry of semi-homogeneous bundles:

• $\mathcal{X}_\alpha \to X$9-twisted sheaves: Defined as equivalence classes $G_m$0 with $G_m$1 and compatibility relations for integral twists. $G_m$2-twists are concretely implemented via pullbacks and powers of ample line bundles (Alvarado et al., 2024, Ito, 2021).
• M-regularity and linear systems: The theory of generic vanishing and M-regularity extends to the $G_m$3-twisted context, yielding new criteria for basepoint-freeness, projective normality, and higher syzygies in terms of the behavior of $G_m$4-twisted ideal sheaves (Ito, 2021, Alvarado et al., 2024).
• Fourier–Mukai transforms: The derived and cohomological behavior of $G_m$5-twisted sheaves is governed by succinct transformation rules under the Fourier–Mukai–Poincaré transform, leading to duality formulas for section modules and obstructions to symmetric generation (Alvarado et al., 2024).

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Twisted coherent sheaves provide the natural framework for describing and manipulating sheaves in the presence of torsion Brauer classes, both on schemes and stacks. Their theory mediates between classical algebraic geometry, derived categories, symplectic geometry, and arithmetic, allowing the realization of new moduli spaces, derived equivalence phenomena, and sophisticated enumerative invariants. Twisted sheaves encode the obstruction to geometric triviality, provide fine control over moduli and deformation, and enable the passage from local-to-global constructions in complicated cohomological backgrounds.