Second Orbifold Chern Class

• The second orbifold Chern class is a cohomological invariant that extends the classic second Chern class to singular, orbifold, and stack contexts using analytic, birational, and algebraic methods.
• It plays a pivotal role in extending geometric inequalities like the Miyaoka–Yau and Bogomolov–Gieseker inequalities to spaces with quotient and mild singularities.
• The invariant provides effective diagnostic tools for characterizing varieties, such as finite torus quotients, and assessing projective flatness and stability in complex geometry.

The second orbifold Chern class is a cohomological invariant generalizing the notion of the standard second Chern class to the setting of complex analytic spaces, stacks, or pairs with orbifold (V-manifold, quotient, or stack) singularities. It provides a crucial ingredient in extending results and inequalities from classical complex and algebraic geometry—such as the Miyaoka–Yau inequality and Bogomolov–Gieseker inequalities—to singular and/or stack-theoretic situations, and delivers effective methods to characterize broad classes of varieties, notably finite quotients of tori and varieties with mild singularities.

1. Formal Definitions and Constructions

Let $X$ be a complex analytic space (or Deligne–Mumford stack) of pure dimension $n$, possibly with klt (Kawamata–log-terminal) or quotient singularities, and $E$ a reflexive coherent sheaf (or vector bundle on a stack). The classical Chern classes $c_i(E)$ are ill-defined globally in the presence of singularities or stack structure. Instead, the orbifold locus $X^\circ \subset X$, defined as the set of points locally analytically isomorphic to quotients of smooth space by finite group actions, admits the structure of a complex orbifold (V-manifold), possibly outside an analytic subset $Z \subset X$ of codimension at least $3$.

Second orbifold Chern class $c_2^{\mathrm{orb}}(E)$ is constructed as follows:

• Analytic and Dolbeault approaches: On $X^\circ$, $E|_{X^\circ}$ is an orbifold vector bundle. Any smooth orbifold Hermitian metric $h$ yields a curvature form $\Theta(E, h)$, and Chern–Weil theory establishes

$$c_2^{\mathrm{orb}}(E, h) = \frac{1}{8\pi^2}\left( \mathrm{Tr}(\Theta \wedge \Theta) - \frac{1}{r}\mathrm{Tr}\Theta \wedge \mathrm{Tr} \Theta \right) \in \mathcal{A}_{\mathrm{orb}}^{2,2}(X^\circ)$$

whose cohomology class in $H_{\mathrm{dR,\,orb}}^4(X^\circ)$ is independent of $h$ (Guenancia et al., 13 Jan 2026).

• Extension to singular $X$: If $\mathrm{codim}_X(X \setminus X^\circ) \geq 3$, Poincaré duality and functorial isomorphisms allow for a well-defined element $$c_2^{\mathrm{orb}}(E) \in H^{2n-4}(X, \mathbb{R})^\vee$$ by lifting the dual of $c_2(E|_{X^\circ})$ (Graf et al., 2017, Ou, 25 Dec 2025).
• Stack-theoretic and algebraic contexts: For $X$ a smooth, proper Deligne–Mumford stack, and $E$ a vector bundle, $c_2^{\mathrm{orb}}(E)$ is defined via the orbifold Chern character in the orbifold Chow ring, incorporating contributions from each sector $I(\Psi)$ of the inertia stack:

$$c_2^{\mathrm{orb}}(E) = \sum_{\Psi}(i_\Psi)_* P_\Psi$$

with $P_\Psi$ an explicit polynomial in the Chern and Todd classes of $E_\Psi$ (the sector restriction) and the logarithmic trace of the tangent bundle (Fu et al., 2018).

2. Functoriality, Birational Variants, and Algebraic Comparisons

The notion of the second orbifold Chern class displays strong functorial properties under orbifold modifications and enjoys desirable stability with respect to birational morphisms that are isomorphisms in codimension at least $3$.

• Birational second Chern class: When the orbifold locus fails to be smooth in codimension $2$, $c_2^{\mathrm{orb}}$ is generally undefined. In these situations, one uses a birational Chern class, constructed via a resolution $f: Y \to X$ minimal in codimension $2$:

$$c_2(X) \cdot \alpha = \int_Y c_2(T_Y) \wedge f^*(\alpha)$$

for all $\alpha \in H^2(X, \mathbb{R})$. This class (for threefolds) distinguishes actions free in codimension $2$ (Graf et al., 2017).

• Comparison with Schwartz–MacPherson Chern classes: For singular spaces, classical SM Chern classes $c_2^{\mathrm{SM}}(X)$, birational $c_2(X)$, and orbifold $c_2^{\mathrm{orb}}(X)$ can differ substantially. If $X$ is smooth in codimension $2$, these notions coincide; otherwise they may disagree and encode sensitive data about singularities and stack structure (Graf et al., 2017).
• Algebraic and analytic approaches: The algebraic construction, e.g. via Mumford’s method (hyperplane sections and orbifold surface reductions), recovers the analytic formulation in the projective case. Cohomological definitions in the analytic category rely on orbifold metrics and currents, together with partial resolutions and Andreotti–Grauert vanishing (Guenancia et al., 13 Jan 2026).

Notion	Well-defined when	Agreement with $c_2^{\mathrm{orb}}$
Birational $c_2(X)$	Resolutions exist, minimal in codim $\geq 2$	If smooth in codim $2$
SM $c_2^{\mathrm{SM}}(X)$	For singular algebraic spaces	If smooth in codim $2$
Stack/Orbifold $c_2^{\mathrm{orb}}$	On $X^\circ$, or globally on [X/G] with finite $G$	Always with classical Chern when smooth

3. Integration, Pairings, and Numerical Invariants

The utility of $c_2^{\mathrm{orb}}$ arises from its capacity to produce well-defined numerical invariants via intersection theory—most critically, through the pairing with powers of Kähler or ample classes. Precisely, for a compact Kähler space $X$, Kähler class $\omega$, and reflexive sheaf $\mathscr E$,

$$c_2(\mathscr E)\cdot \{\omega\}^{n-2} = \int_{X \setminus Z} \frac{1}{8\pi^2} \left(\mathrm{Tr}(\Theta \wedge \Theta) - \frac{1}{r} \mathrm{Tr}\Theta \wedge \mathrm{Tr}\Theta\right) \wedge \widehat\Omega$$

with $\widehat\Omega$ a compactly supported form on $X \setminus Z$, independent of the metric or decomposition up to exact forms (Guenancia et al., 13 Jan 2026).

On Deligne–Mumford stacks, the Chow-level orbifold Chern character enables extraction of concrete, sectorwise corrections to the Chern numbers, generalizing the Satake–Thurston formalism to higher K-theory and motivic cohomology settings (Fu et al., 2018).

4. Vanishing Criteria, Flatness, and Characterization of Convex Structures

The second orbifold Chern class serves as a decisive diagnostic for the birational or uniformization type of $X$ in the following settings:

• Torus quotient characterization: For compact complex threefolds with canonical singularities, the vanishing of $c_1(X)$ together with the vanishing of $c_2^{\mathrm{orb}}(X)\cdot \omega$ for some Kähler class $\omega$ is both necessary and sufficient for $X$ to be a finite quotient of a three-dimensional complex torus by a group acting freely in codimension $1$ (Graf et al., 2017, Guenancia et al., 13 Jan 2026). When $c_2^{\mathrm{orb}}$ is not defined (actions free only in codim $2$), the birational second Chern class is employed.
• Flatness and projectivity: If a reflexive sheaf $\mathscr E$ is $\omega$-stable and the Bogomolov–Gieseker inequality achieves equality, then $\mathscr E$ must be projectively flat on the orbifold locus $X \setminus Z$, and its second orbifold Chern class vanishes (Guenancia et al., 13 Jan 2026).

5. Bogomolov–Gieseker and Miyaoka–Yau-type Inequalities

With $c_2^{\mathrm{orb}}$ in place, generalizations of classical stability inequalities are established for varieties and pairs with quotient or mild singularities.

• Orbifold Bogomolov–Gieseker inequality: If $X$ is a compact Kähler orbifold or mild singularity variety, and $\mathscr E$ (or more generally a reflexive $\mathbb{Q}$-sheaf) is $\omega$-slope stable, one has (Ou, 25 Dec 2025, Guenancia et al., 13 Jan 2026):

$$\left(2r\,c_2(\mathscr E) - (r-1)\,c_1(\mathscr E)^2\right) \cdot [\omega]^{n-2} \geq 0$$

with equality precisely for projectively flat orbifold bundles.

• Pseudo-effectivity and Miyaoka–Yau extensions: For threefold pairs or log pairs $(X, D)$ with $K_X + D$ in the movable cone and mild singularities, $c_2^{\mathrm{orb}}(X, D) \cdot A \geq 0$ for all ample divisors $A$, generalizing Miyaoka’s pseudo-effectivity of $c_2$ (Rousseau et al., 2016). The inequality specializes to

$(K_X + D)^2 \cdot A \leq 3\,c_2(X, D) \cdot A$

for ample $A$.

6. Computational Techniques and Examples

Explicit computations of $c_2^{\mathrm{orb}}$ in stacks or global quotient orbifolds exploit sector decompositions:

1. Inertia stack approach: Decompose the inertia stack $IX = \bigsqcup_{\Psi} I(\Psi)$, compute the Chern and Todd classes on each sector, and sum their contributions as prescribed by the orbifold Chern character formalism (Fu et al., 2018).
2. Resolution-based methods: Use adapted (Kawamata) covers or functorial resolutions to pull back and compute Chern classes where the bundle becomes locally free (Rousseau et al., 2016, Ou, 25 Dec 2025).
3. Illustrative calculations: For $X = (T^2) /\{\pm1\}$ (Kummer surface), the orbifold $c_2^{\mathrm{orb}}(X) = 0$, the birational $c_2(X) = 24$, and the Schwartz–MacPherson class $c_2^{\mathrm{SM}}(X) = 8$, reflecting differing sensitivity to singularities and group action types (Graf et al., 2017).

7. Applications in Geometry and Classification Theory

The second orbifold Chern class undergirds several foundational results:

• Classification of torus quotient threefolds and characterization of their uniformization by vanishing conditions of Chern classes (Graf et al., 2017, Guenancia et al., 13 Jan 2026).
• Proofs of effectivity and finiteness statements for varieties of general type, e.g., effective non-vanishing theorems and boundedness of non-general-type subvarieties (Rousseau et al., 2016).
• Facilitation of general orbifold versions of classical theorems such as Donaldson–Uhlenbeck–Yau on Hermite–Einstein metrics and their implications for stability and moduli (Ou, 25 Dec 2025).

The second orbifold Chern class thus provides a flexible, robust framework for extending the reach of intersection-theoretic, stability, and classification results into singular, stacky, or orbifold geometric contexts.