Bulk-Cone Singularities in Complex Geometry

• Bulk-cone singularities are defined as interior conical structures in manifolds and varieties, detectable through analytic and algebraic methods.
• Their study employs degenerate Monge–Ampère equations and robust regularization techniques to establish existence and rigidity results.
• Applications range from Kähler–Einstein metrics and moduli compactifications to tensor field theories, driving advances in complex geometry.

Bulk-cone singularities are a class of geometric, analytic, and algebraic singularities characterized by their arising either from conical structures in the bulk of a manifold or variety (as opposed to on the boundary), or as singular features in physical or mathematical models detectable through bulk geodesics, intersection theory, or nonlinear analysis. They appear in several domains, notably complex differential geometry, algebraic and Kähler geometry, moduli theory, microlocal analysis, and mathematical physics. The following sections synthesize the definitions, analytic framework, existence and rigidity results, applications (including to tensor field theory and moduli problems), and open issues related to bulk-cone singularities, with particular emphasis on their appearance in Kähler-Einstein geometry with cone singularities.

1. Definitions and Local Models

A prototypical example of bulk-cone singularities arises in Kähler geometry as metrics with cone singularities along divisors. Let $X$ be a compact complex manifold and $D$ a divisor with simple normal crossing support:

$D = \sum_{j\in J}(1-\tau_j)Y_j\,,$

where each $Y_j$ is a smooth hypersurface, and $0 < \tau_j \leq 1$. The canonical local model for a Kähler metric with cone angles $2\pi\tau_j$ along $Y_j$ is given in adapted coordinates $(z_1,\ldots, z_n)$ (with $Y_j = \{z_j=0\}$ for $j=1,\dots,d$) by

$$\omega_0 = \sqrt{-1} \sum_{j=1}^d \frac{dz_j \wedge d\overline{z}_j}{|z_j|^{2(1-\tau_j)}} + \sqrt{-1} \sum_{k=d+1}^n dz_k \wedge d\overline{z}_k\,.$$

A metric $\omega$ on $X$ is said to have cone singularities along $D$ if it is quasi-isometric to $\omega_0$ off $D$. In the broader sense, bulk-cone singularities refer to isolated or non-isolated singularities in the interior (“bulk”) of a space resulting from a conical local metric or structure, not merely due to boundary or divisorial data.

In moduli theory and algebraic geometry, analogous bulk-cone singularities arise from the combinatorics of rational polyhedral cone decompositions (e.g., in the theory of toroidal compactifications of moduli spaces) and are linked to the failure of certain cones to be "basic" or "simplicial" (see Section 3).

2. Analytic Framework: Monge–Ampère Equations and Regularization

The analytic paper of bulk-cone singularities in Kähler-Einstein metrics proceeds via the construction of solutions to degenerate Monge–Ampère equations with singular right-hand sides encoding the conical data:

$$(\omega_\varepsilon + dd^c \varphi_\varepsilon)^n = \frac{e^{f + \lambda(\varphi_\varepsilon + \psi_\varepsilon)}}{\prod_{j\in J} (\varepsilon^2 + |s_j|^2)^{1-\tau_j}} \omega^n\,,$$

where the regularization parameter $\varepsilon \to 0$, $s_j$ are local equations for $Y_j$, and $f,\psi_\varepsilon$ encode potential and curvature normalization.

Under the small cone angle condition $\tau_j\leq 1/2$ for all $j$, the approximate metrics produced by this regularization procedure satisfy uniform lower bounds on holomorphic bisectional curvature. This uniformity is critical for applying Kołodziej’s $\mathcal{C}^0$ estimates, Evans–Krylov higher order theory, and the Laplacian comparison techniques required for passage to the limit and ensuring quasi-isometry to the local model (see (Campana et al., 2011, Borbon, 2016)).

In the positive curvature (Fano/log-Fano) case, such as $-(K_X+D)$ ample, the analytic treatment is subtler due to the absence of uniform curvature control. The existence of solutions typically relies on variational methods or continuity paths, and regularization arguments only guarantee the cone singularity structure if a weak solution with bounded potential is known a priori.

3. Geometric and Algebraic Manifestations

3.1. Moduli and Toroidal Compactifications

In the paper of moduli spaces of abelian varieties and related spaces, "bulk-cone singularities" in the stack or compactification correspond to nonbasic or non-simplicial cones in rational polyhedral decompositions. For example, in the perfect cone (first Voronoi) compactification of the Siegel moduli space $\mathcal{A}_g^{\operatorname{Perf}}$:

• A unique 10-dimensional cone associated to the $D_4$ root lattice is not basic or simplicial, causing the singular locus to have codimension 10 for $g\geq4$;
• In the second Voronoi compactification, non-simplicial cones of dimension 3 lead to singularities in codimension 3 for $g\geq5$ (Sikirić et al., 2013).

3.2. Algebraic Boundedness

In algebraic geometry, singularities described as "cones" (normal affine varieties with isolated fixed points under a torus action) are shown to form bounded families under dimension, $\epsilon$-log canonical threshold, and isotropy bounds. This has implications for the birational classification and invariants such as minimal log discrepancies and fundamental groups, especially in the context of families of degenerations or Gromov–Hausdorff limits (Moraga, 2018).

4. Kähler–Einstein Metrics with Bulk-Cone Singularities

For a compact Kähler manifold $X$ and a simple normal crossing divisor $D$, the existence of Kähler–Einstein metrics with cone singularities along $D$ is established under key conditions:

• In negative or zero Ricci curvature regimes ($K_X+D$ ample or numerically trivial), existence is guaranteed for $0<\tau_j\leq1/2$ via regularization and sharp analytic estimates (Campana et al., 2011).
• In positive curvature, existence requires weaker assumptions, with the proof that any weak KE metric (with bounded potential) must have prescribed cone singularities.
• For klt pairs $(X, D)$ with $K_X+D$ big or $-(K_X+D)$ ample and all coefficients of $D$ in $[\frac12,1)$, the KE metric attached to $(X, D)$ exhibits cone singularities on the log-smooth locus and the ample locus of $K_X+D$, unifying the theory for both general type and Fano/log-Fano settings (Guenancia, 2012).

Locally, the metric is quasi-isometric to a standard cone model:

$$\omega_{\mathrm{mod}} = \sum_{j=1}^k \frac{i\,dz_j\wedge d\overline{z}_j}{|z_j|^{2(1-\beta_j)}} + \sum_{j=k+1}^n i\, dz_j \wedge d\overline{z}_j\,,$$

exhibiting controlled asymptotics and reflecting "bulk-cone" geometry.

5. Applications to Holomorphic Tensor Fields

An important application is the extension of Lichnerowicz-Kobayashi vanishing and parallelism results to orbifolds and spaces with cone singularities:

• Definition of orbifold tensor bundles $T_s^r(X|D)$ extends the framework of holomorphic sections to cone-orbifold settings;
• Boundedness of holomorphic tensors (in the sense that $|u(x)|_g$ remains bounded up to $D$) ensures extendibility as global holomorphic sections;
• For negative $c_1(K_X + D)$, $H^0(X, T^r_s(X|D))=0$ for $r>s$ (with a strengthened numerical inequality), generalizing the classical Bochner vanishing to this singular setting;
• For vanishing or positive $c_1(K_X + D)$, parallelism or vanishing of symmetric tensors holds;
• The analytic core is a Bochner-type formula on $X_0$ with careful cut-off arguments to handle the singularity, involving a musical isomorphism ("#-operator") which preserves norm control under the cone metric;
• Applications include constraints on automorphism groups and invariants of the underlying singular spaces (Campana et al., 2011).

6. Technical Framework and Regularity

The mathematical toolkit supporting these results includes:

• Degenerate complex Monge–Ampère theory with $L^p$ right-hand sides;
• Kołodziej’s $\mathcal{C}^0$ and Evans–Krylov–Schauder estimates for higher regularity away from $D$;
• Local definition of singular tensor bundles incorporating conic monodromy and orbifold corrections;
• Use of Laplacian and curvature estimates uniform in the cone angle parameters;
• Regularization and bootstrapping arguments essential for passing from smooth to conic metrics.

The framework is robust enough to treat variations such as:

• Orbifold points and non-simple divisors (by local finite covers or index computations);
• Log-Fano or log-Calabi–Yau cases via adapted normalization and variational solution spaces.

7. Significance, Extensions, and Open Problems

Bulk-cone singularities present both rigidity and flexibility: rigidity in the sense that analytic and geometric estimates pin down the asymptotic structure, flexibility by allowing for varying parameters (e.g., cone angles) and accommodating extensions to the general type, Fano, and Calabi–Yau regimes.

Extensions and open questions include:

• Behavior and regularity up to loci where the cone class is merely big and not nef, or near more severe singularities of the underlying space (Guenancia, 2012);
• Modular and automorphic implications for the classification and enumeration of such metrics, including applications to compactifications and structural results for moduli spaces (Sikirić et al., 2013);
• Further analytic regularity, such as full control up to the divisor or boundary of the ample locus, and existence theory for more singular or minimal models;
• Stability considerations and deformations, especially for tangent cone or “bulk” limit models, in both algebraic and differential geometric settings (Borbon, 2015).

The precise asymptotic and rigidity properties of bulk-cone singularities are essential to moduli theory, the minimal model program, degenerations in Kähler geometry, and the general paper of canonical metrics and their singularities. The analytic, algebraic, and topological interplay captured by these results continues to inform contemporary research in complex geometry and its interactions with algebraic and arithmetic geometry.